一种Polar码编码方法及装置与流程

本申请实施例涉及通信技术领域，尤其涉及一种polar码编码方法及装置。

背景技术：

信道编码作为最基本的无线接入技术，在保证数据的可靠性传输方面起到至关重要的作用。在现有的无线通信系统中，一般采用turbo码、低密度奇偶校验码(lowdensityparitycheck，ldpc)和极化(polar)码进行信道编码。turbo码不能够支持过低或过高码率的信息传输。而对于中短包传输，turbo码和ldpc码也由于自身编译码的特点，在有限码长下很难达到理想的性能。在实现方面，turbo码和ldpc码在编译码实现过程中具有较高的计算复杂度。polar码是理论上证明可以取得香农容量，且具有相对简单的编译码复杂度的好码，因而得到了越来越广泛的应用。

但是，随着无线通信系统的快速演进，第五代(5thgeneration，5g)通信系统等未来的通信系统将会出现一些新的特点。例如，最典型的三个通信场景包括增强型移动互联网(enhancemobilebroadband，embb)、海量机器连接通信(massivemachinetypecommunication，mmtc)和高可靠低延迟通信(ultrareliablelowlatencycommunication，urllc)。这些通信场景对于polar码的编译码性能提出了更高的要求。

极化信道的可靠度排序对polar码的编译码性能起到重要作用，而现阶段，极化信道的可靠度排序的准确度并不理想，从而影响了polar码在应用过程中的编译码性能的进一步提高。

技术实现要素：

本申请实施例提供一种polar码编码方法及装置，用以提高极化信道的可靠度排序的准确度。

第一方面，提供一种polar码编码方法，采用逐比特，逐母码长度的构造方法，获得polar码构造的排序序列，更能反映信道的实际可靠度，有助于提高极化信道的可靠度排序的准确度，改善polar码的编译码性能。

在一个可能的设计中，确定用于对待编码比特进行编码的排序序列，所述排序序列用于表征n个极化信道的可靠度排序，采用所述排序序列对所述待编码比特进行polar码编码，得到编码后的比特，其中，n为polar码的母码长度，n为2的正整数次幂。

在一个可能的设计中，所述待编码比特长度为k，k个所述待编码比特被映射到所述n个极化信道中的k个极化信道上，所述k个极化信道的可靠度高于剩余的n-k个极化信道的可靠度。

在一个可能的设计中，若所述排序序列按照极化信道的可靠度由低到高的顺序排列，则通过以下方法确定本申请的用于对待编码比特进行编码的排序序列：首先，获取第一序列，第二序列，第三序列，第四序列和第五序列；然后，获取所述第二序列中序号小于等于64的第一子序列，按照所述第一子序列的前(64-t1+1)个序号在所述第一序列中的排序，更新所述第一子序列的前(64-t1+1)个序号在所述第一子序列中的排序，基于更新后的第一子序列获得待对比第二序列；基于第一目标长度k1，分别在所述第二序列和所述待对比第二序列中选择k1个最可靠位置对应的序号k1个最可靠位置对应的序号，分别基于选择的序号和已生成的第二中间结果序列获得编码码字性能结果，其中，所述第二中间结果序列中包含已选择的(k1-1)个序号，所述目标长度k1依次取值从t1到kmax^(m)，kmax^(m)为母码长度为m时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新所述第二中间结果序列；将k1取值为kmax^(m)时获得的更新后的第二中间结果序列作为更新后第一子序列，将所述更新后第一子序列置于所述第二序列中，获得更新后第二序列；接着，获取所述第三序列中序号小于等于128的第二子序列，按照所述第二子序列的前(128-t2+1)个序号在所述更新后第二序列中的排序，更新所述第二子序列的前(128-t2+1)个序号在所述第二子序列中的排序，基于更新后的第二子序列获得待对比第三序列；基于第二目标长度k2，分别在所述第三序列和所述待对比第三序列中选择k2个最可靠位置对应的序号k2个最可靠位置对应的序号，分别基于选择的序号和已生成的第三中间结果序列获得编码码字性能结果，其中，所述第三中间结果序列中包含已选择的(k2-1)个序号，所述第二目标长度k2依次取值从t2到kmax^(m)，kmax^(m)为母码长度为m时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新所述第三中间结果序列；将k2取值为kmax^(m)时获得的更新后的第三中间结果序列作为更新后第二子序列，将所述更新后第二子序列置于所述第三序列中，获得更新后第三序列；接着，获取所述第四序列中序号小于等于256的第三子序列，按照所述第三子序列的前(256-t3+1)个序号在所述更新后第三序列中的排序，更新所述第三子序列的前(256-t3+1)个序号在所述第三子序列中的排序，基于更新后的第三子序列获得待对比第四序列；基于第三目标长度k3，分别在所述第四序列和所述待对比第四序列中选择k3个最可靠位置对应的序号k3个最可靠位置对应的序号，分别基于选择的序号和已生成的第四中间结果序列获得编码码字性能结果，其中，所述第四中间结果序列中包含已选择的(k3-1)个序号，所述第三目标长度k3依次取值从t3到kmax^(m)，kmax^(m)为母码长度为m时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新所述第四中间结果序列；将k3取值为kmax^(m)时获得的更新后的第四中间结果序列作为更新后第三子序列，将所述更新后第三子序列置于所述第四序列中，获得更新后第四序列；最后，获取所述第五序列中序号小于等于512的第四子序列，按照所述第四子序列的前(512-t4+1)个序号在所述更新后第四序列中的排序，更新所述第四子序列的前(512-t4+1)个序号在所述第四子序列中的排序，基于更新后的第四子序列获得待对比第五序列；基于第四目标长度k4，分别在所述第五序列和所述待对比第五序列中选择k4个最可靠位置对应的序号k4个最可靠位置对应的序号，分别基于选择的序号和已生成的第五中间结果序列获得编码码字性能结果，其中，所述第五中间结果序列中包含已选择的(k4-1)个序号，所述第四目标长度k4依次取值从t4到kmax^(m)，kmax^(m)为母码长度为m时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新所述第五中间结果序列；将k4取值为kmax^(m)时获得的更新后的第五中间结果序列作为更新后第四子序列，将所述更新后第四子序列置于所述第五序列中，获得更新后第五序列；其中，t1、t2、t3和t4均为正整数，包括最小信息比特数加crc比特及pc比特数。

在一个可能的设计中，t1、t2、t3和t4由所述性能优劣判断规则确定。

在一个可能的设计中，可以通过以下方法来获取第一序列，第二序列，第三序列，第四序列和第五序列：根据第五目标长度k5，在(m-k5+1)个序号中依次选择一个序号，每选择一个序号，基于所选择的一个序号与已生成的中间结果序列获得编码码字性能结果，其中，所述中间结果序列包含已选择的(k5-1)个序号，所述(m-k5+1)个序号与所述中间结果序列不相交，所述目标长度k5依次取值从1到kmax^(m)，kmax^(m)为母码长度为m时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新所述中间结果序列；其中，m为2的整数次幂，在m＝64时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为所述第一序列；在m＝128时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为所述第二序列；在m＝256时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为所述第三序列；在m＝512时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为所述第四序列；在m＝1024时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为所述第五序列。

在一个可能的设计中，在n＝1024时，所述排序序列可以为说明书中序列序列①至序列中的任意一个序列的部分或者全部。

通过所述排序序列进行polar构造，可以有助于提高对极化信道可靠度的评估准确性，从而提升polar码的编译码性能。

说明书中序列①至序列中的任意一个序列中的序号最小值是1，且按照n个极化信道的可靠度从低到高排列的。在一个可能的设计中，若所述排序序列按照所述n个极化信道的可靠度从低到高排列的，且所述排序序列的序号最小值为0，则将任意一个序列中的每个序号均减1即可得到新的序列，序列性能并不会受到影响。

类似的，在一个可能的设计中，若所述排序序列按照所述n个极化信道的可靠度从高到低排列的，则将任意一个序列中的序号反向排列即可得到新的序列，序列性能同样不会受到影响。

在一个可能的设计中，所述排序序列还可以利用各个信道的归一化可靠度或等效可靠度序列进行表征。例如：信道x在上述序列的排序位置为n(排序序列最小序号记为1)，则该信道的可靠度可以表示为n或者归一化的n/n。

在一个可能的设计中，所述排序序列中少数元素之间的位置可以互换。例如，序号位置可以在设定幅度内调整，例如，设定幅度为5，将序号为10的元素位置在左右5个位置内调整均可。

第二方面，提供一种polar码编码装置，该装置具有实现上述第一方面和第一方面的任一种可能的设计中所述的方法的功能。所述功能可以通过硬件实现，也可以通过硬件执行相应的软件实现。所述硬件或软件包括一个或多个与上述功能相对应的模块。

在一个可能的设计中，当所述功能的部分或全部通过硬件实现时，所述polar码编码装置包括：输入接口电路，用于获取待编码比特；逻辑电路，用于执行上述第一方面和第一方面的任一种可能的设计中所述的行为；输出接口电路，用于输出编码后的比特序列。

可选的，所述polar码编码装置可以是芯片或者集成电路。

在一个可能的设计中，当所述功能的部分或全部通过软件实现时，所述polar码编码装置包括：存储器，用于存储程序；处理器，用于执行所述存储器存储的所述程序，当所述程序被执行时，所述polar码编码装置可以实现如上述第一方面和第一方面的任一种可能的设计中所述的方法。

可选的，上述存储器可以是物理上独立的单元，也可以与处理器集成在一起。

在一个可能的设计中，当所述功能的部分或全部通过软件实现时，所述polar码编码装置包括处理器。用于存储程序的存储器位于所述编码装置之外，处理器通过电路/电线与存储器连接，用于读取并执行所述存储器中存储的程序。

第三方面，提供了一种通信系统，该通信系统包括发送端和接收端，所述发送端可以执行如上述第一方面及其可能的设计所述的方法。

第四方面，提供了一种计算机存储介质，存储有计算机程序，该计算机程序包括用于执行第一方面和第一方面的任一可能设计中任一种所述的方法的指令。

第五方面，本申请实施例提供了一种包含指令的计算机程序产品，当其在计算机上运行时，使得计算机执行上述各方面所述的方法。

附图说明

图1为本申请实施例中应用的通信系统架构示意图；

图2为本申请实施例中polar码编码方法的流程示意图；

图3为本申请实施例中polar码编码装置结构示意图之一；

图4为本申请实施例中polar码编码装置结构示意图之二；

图5为本申请实施例中polar码编码装置结构示意图之三；

图6为本申请实施例中polar码编码装置结构示意图之四。

具体实施方式

下面将结合附图，对本申请实施例进行详细描述。

本申请实施例提供一种polar码编码方法及装置，根据polar码原理，极化操作对各极化信道的可靠度产生影响，使得它们有的可靠度高，有的低，选择可靠度高的极化信道用于传输信息比特，则可实现好的传输性能。本申请实施例从polar码基本原理出发，采用逐比特，逐母码长度的构造方法，获得polar码构造的排序序列，更能反映信道的实际可靠度，有助于提高极化信道的可靠度排序的准确度，改善polar码的编译码性能。

为方便对本申请实施例的理解，下面对polar码作简单介绍。

polar码的编码策略利用无噪信道传输用户有用的信息，全噪信道传输约定的信息或者不传信息。polar码也是一种线性块码，其编码矩阵为gn，编码过程为其中是一个二进制的行矢量，长度为n(即码长)；gn是一个n×n的矩阵，且定义为log2n个矩阵f2的克罗内克(kronecker)乘积。上述矩阵

polar码的编码过程中，中的一部分比特用来携带信息，称为信息比特集合，这些比特的索引的集合记作另外的一部分比特设置为接收端和发送端预先约定的固定值，称之为固定比特集合或冻结比特集合(frozenbits)，其索引的集合用的补集表示。polar码的编码过程相当于：这里，gn(a)是gn中由集合中的索引对应的那些行得到的子矩阵，gn(ac)是gn中由集合中的索引对应的那些行得到的子矩阵。为中的信息比特集合，数量为k；为中的固定比特集合，其数量为(n-k)，是已知比特。这些固定比特通常被设置为0，但是只要接收端和发送端预先约定，固定比特可以被任意设置。从而，polar码的编码输出可简化为：这里为中的信息比特集合，为长度k的行矢量，即|·|表示集合中元素的个数，k为信息块大小，是矩阵gn中由集合中的索引对应的那些行得到的子矩阵，是一个k×n的矩阵。

polar码的构造过程即集合的选取过程，决定了polar码的性能。polar码的构造过程通常是，根据母码码长n确定共存在n个极化信道，分别对应编码矩阵的n个行，计算极化信道可靠度，将可靠度较高的前k个极化信道的索引作为集合的元素，剩余(n-k)个极化信道对应的索引作为固定比特的索引集合的元素。集合决定了信息比特的位置，集合决定了固定比特的位置。

本申请实施例涉及的排序序列，用于表征n个极化信道的可靠度排序，n为polar码的母码长度，n为2的正整数次幂。

一种情况下，该排序序列中可以包括n个极化信道的序号，n个极化信道的序号在排序序列中是按照n个极化信道的可靠度排列的，例如，可以按照可靠度由高到底的顺序排列，也可以按照可靠度由低到高的顺序排列，为方便说明，将这种排序序列称为q序列。

另一种情况下，该排序序列中可以包括n个极化信道的可靠度的值，可选的，该可靠度的值可以为归一化的可靠度的值或者等效可靠度的值，例如，例如：信道x在排序序列的排序位置为n(排序序列中最低序号记为1)，则该信道的可靠度可以表示为n或者归一化的n/n。类似的，n个极化信道的可靠度的值在排序序列中可以按照可靠度由高到底的顺序排列，也可以按照可靠度由低到高的顺序排列，为方便说明，将这种排序序列称为z序列。可知，q序列和z序列可以互相转化，即已知q序列，可以求得z序列，反之亦然。

该排序序列中元素的取值可以从0～n-1，也可以从1～n。本申请实施例中，以排序序列中元素取值为1～n为例进行说明，可知，将排序序列中每个元素的值依次减去1可以获得元素取值为0～n-1的排序序列，两种排序序列的性能相同，均可表征n个极化信道可靠度排序。本申请实施例设计的排序序列，以按照可靠度由低到高的顺序排列为例进行说明，可以理解，将排序序列的每个元素逆序排列可以获得按照可靠度由高到低的顺序排列的排序序列，两种排序序列的性能相同，均可表征n个极化信道可靠度排序。

如图1所示，本申请实施例应用的通信系统100中包括发送端101和接收端102。发送端101也可以称为编码端，接收端102也可以称为译码端。其中，发送端101可以为基站，接收端102为终端；或者，发送端101为终端，接收端102为基站。基站是一种部署在无线接入网中用以为终端提供无线通信功能的装置。基站可以包括各种形式的宏基站，微基站，中继站，接入点等等。可以应用在不同的无线接入技术的系统中，例如长期演进(longtermevolution，lte)系统中，或者，第五代(5thgeneration，5g)通信系统等更多可能的通信系统中。基站还可以是其他具有基站功能的网络设备，特别地，还可以是d2d通信中担任基站功能的终端。终端可以包括各种具有无线通信功能的手持设备、车载设备、可穿戴设备、计算设备或连接到无线调制解调器的其他处理设备，以及各种形式的用户设备(userequipment，ue)，移动台(mobilestation，ms)等。

基于图1所示的通信系统架构，本申请实施例中，执行polar码编码方法的执行主体可以为发送端101。下面将对本申请实施例提供的polar码编码方法做详细介绍。

基于图1所示的通信系统架构，如图2所示，本申请实施例提供的polar码编码方法具体流程如下所述。

步骤201、确定用于对待编码比特进行编码的排序序列。

其中，该排序序列用于表征n个极化信道的可靠度排序，n为polar码的母码长度，n为2的正整数次幂。

步骤202、采用该排序序列对待编码比特进行polar码编码，得到编码后的比特。

其中，若待编码比特长度为k，k个待编码比特被映射到n个极化信道中的k个极化信道上，k个极化信道的可靠度高于剩余的n-k个极化信道的可靠度。

实际应用中，可将此排序序列离线存储，给定传输参数(例如码长和码率)后，利用预先存储的排序序列来确定polar的构造方法，即确定构造polar码所用的信息比特集合和冻结比特集合，例如，可以通过读表的方式获得polar码的构造方法。

下面具体介绍一下如何获得本申请实施例应用的排序序列。

首先说明的是，本申请在获得排序序列过程中，会应用到比较序列性能优劣的规则，本申请实施例给出一种性能优劣判断规则的举例，但并不限于此。

排序序列需要满足简单的嵌套特征，即当最大母码长度的排序序列确定后，可以根据最大母码码长的排序序列获得其它较短母码长度的排序序列。一种可能的性能优劣判断规如下所述。

评估度量：误块率(blockerrorrate，bler)达到10^-2和10^-3所需要的信噪比(signal-to-noiseratio，snr)；

仿真假设：

分别评估bler达到10^-2和10^-3所需要的snr值。即给定序列a和b，按上表所述仿真参数，考察各(n,k)组合，例如对于n＝64，k＝10，使用序列a的polar码达到bler＝10^-2所需snr值记为ha，使用序列b的polar达到bler＝10^-2所需snr值记为hb，将它们的插值ha-hb与给定门限进行比较，大于门限值则记为序列b获胜，序列b的获胜数量增加1，序列a的失败数量增加1；小于负的门限值，则记为序列a获胜，序列a的获胜数量增加1，序列b的失败数量增加1；小于等于门限且大于等于负的门限时，记为两个序列性能相同，它们的获胜数量都不增加。进一步考察序列a和b在达到bler＝10^-3时所需的snr值，将插值和门限做比较，并对获胜数量做进一步操作。

上述门限的确定步骤如下：

待编码信息长度k小于等于200时，记perfthresh_k＝0.1db；k大于200时perfthresh_k＝0.3db；

译码列表数l＝1时，perfthresh_l＝0.4db；l＝2时，perfthresh_l＝0.2db；l为其他值时，perfthresh_l＝0.1db

最终门限取上述两个门限值中较大的一个，即perfthresh＝max(perfthresh_k,perfthresh_l)。

具体序列对比流程如下：

step1：计算各序列的获胜数量，失败数量，整体失败数量。取序列a,与序列b在每一个仿真案例上进行比较，对于每一个仿真案例：如果a的性能比b差perfthresh,则a对b的失败数量failcount_ab加1，如果a的性能比b好perfthresh,则a对b的获胜数量wincountab加1，如果(failcount_ab–wincountab)/总仿真案例数量>2％,则a的整体失败数量overallfail_a加1，与序列c在每一个仿真案例上进行比较。对于每一个仿真案例：如果a的性能比c差perfthresh,则a对c的失败数量failcount_ac加1，如果a的性能比c好perfthresh,则a对c的获胜数量wincount_ac加1，如果(failcount_ac–wincountac)/总仿真案例数量>2％,则a的整体失败数量overallfail_a加1。对于序列d到n重复该过程，取序列b，与序列b在每一个仿真案例上进行比较，以此类推。取序列n,与序列a进行比较，以此类推。选择整体失败数量overallfail最小的序列。

step2，如果多个序列a到m都有最小的整体失败数量overallfail,则对于序列a,和序列b在每一个仿真案例上进行比较，对于每一个仿真案例，如果a的性能比b好perfthresh,a对b的获胜数量wincount_ab加1，和序列c在每一个仿真案例上进行比较。对于每一个仿真案例，如果a的性能c好perfthresh,a对c的获胜数量wincount_ac加1，a的总体获胜数量wincounta＝∑wincount_ab…am，重复该过程直到m。对于b到m序列重复该过程，选取总体获胜数量wincount最高的序列,假定该序列为序列w。

step3：给定整体获胜数量最高的序列w，如果存在某序列x，x对w的获胜数量大于w对x的获胜数量，即wincount_xw>wincountwx,那么序列x也会被选择进入下一轮。

step4：如果多个序列被选择，那么最终序列通过投票来确定。

以排序序列为q序列为例，本申请实施例应用的排序序列的获得方法大致分为两个步骤：

一、依次获得母码码长为64、128、256、512、1024时，按照性能优劣判断规则获得的性能最优的序列。

为方便描述，母码码长为64时的序列记为第一序列或q64，母码码长为128时的序列记为第二序列或q128，母码码长为256时的序列记为第三序列或q256，母码码长为512时的序列记为第四序列或q512，母码码长为1024时的序列记为第五序列或q1024。

二、将五个序列进行合并，获得长度为1024的更新序列，作为本申请实施例应用的排序序列。

首先，获取第二序列中序号小于等于64的第一子序列，即q128中元素的值小于等于64的集合。确定第一子序列中排序靠前的前(64-t1+1)个序号集合，记为p64-t1+1。按照第一子序列的前(64-t1+1)个序号在第一序列中的排序，更新第一子序列的前(64-t1+1)个序号在第一子序列中的排序，基于更新后的第一子序列获得待对比第二序列。即，在q64中找到p64-t1+1中的元素的值的排序，利用这个排序顺序替换第一子序列中前(64-t1+1)个序号在第一子序列中的排序顺序，得到q128*，q128*可以称为待对比第二序列。t1为正整数，包括但不限于最小信息比特数加循环冗余校验(cyclicredundancycheck，crc)比特及奇偶校验(paritycheck，pc)比特数，例如，t1通过上述性能优劣判断规则给出的信息比特数和crc长度为例，t1＝8+19＝27，8指示信息比特数，19指示crc长度。基于第一目标长度k1，分别在第二序列和待对比第二序列中选择k1个最可靠位置对应的序号，分别基于选择的序号和已生成的第二中间结果序列获得编码码字性能结果，其中，第二中间结果序列中包含已选择的(k1-1)个序号，目标长度k1依次取值从t1到kmax^(m)，kmax^(m)为母码长度为m＝64时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新第二中间结果序列。将k1取值为kmax^(m)时获得的更新后的第二中间结果序列作为更新后第一子序列，将更新后第一子序列置于第二序列中，获得更新后第二序列，更新后第二序列也可以记为q128^opt。也就是说，从k1＝27开始，到64结束，每取一个k1的值，对比按照q128和q128*分别构造的polar码的性能，即(m＝128，k1)的polar码的性能，m为码长，k1为信息比特序号集合长度，可以理解，此时考察的(m＝128，k1)的polar码的性能也考虑到了(m＝64，k1)的polar码的性能。每选好一个性能较好的序号，依次放入q128^opt中，当q128和q128*两个序列构造的polar码的性能相同时，直接将这些序号放入q128^opt中。

接着，类似的，获取第三序列中序号小于等于128的第二子序列，按照第二子序列的前(128-t2+1)个序号在更新后第二序列中的排序，更新第二子序列的前(128-t2+1)个序号在第二子序列中的排序，基于更新后的第二子序列获得待对比第三序列；基于第二目标长度k2，分别在第三序列和待对比第三序列中选择k2个最可靠位置对应的序号，分别基于选择的序号和已生成的第三中间结果序列获得编码码字性能结果，其中，第三中间结果序列中包含已选择的(k2-1)个序号，第二目标长度k2依次取值从t2到kmax^(m)，kmax^(m)为母码长度为m＝128时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新第三中间结果序列；将k2取值为kmax^(m)时获得的更新后的第三中间结果序列作为更新后第二子序列，将更新后第二子序列置于第三序列中，获得更新后第三序列。通俗的说，取q128^opt和q256，将q256中小于等于128的前(128-t2+1)个的元素集合记为p128-t2+1，在q128^opt中找到p128-t2+1所含元素的顺序，按照这个顺序替换第二子序列中前(128-t2+1)个序号在第二子序列中的顺序，得到长为256的序列q256*。t2为正整数，包括最小信息比特数加crc比特及pc比特数，例如，t2通过上述性能优劣判断规则给出的信息比特数和crc长度为例，t2＝8+19＝27。从k2＝t2开始，到128结束，每取一个k2的值，对比按照q256和q256*构造的polar码的性能，即(m＝256，k2)的polar码的性能，m为码长，k2为信息比特序号集合长度，可以理解，此时考察的(m＝256，k2)的polar码的性能也考虑到了(m＝128，k2)的polar码的性能。每选好一个性能较好的序号，依次放入q256^opt中，当q256和q256*两个序列构造的polar码的性能相同时，直接将这些序号放入q256^opt中。

接着，类似的，获取第四序列中序号小于等于256的第三子序列，按照第三子序列的前(256-t3+1)个序号在更新后第三序列中的排序，更新第三子序列的前(256-t3+1)个序号在第三子序列中的排序，基于更新后的第三子序列获得待对比第四序列；基于第三目标长度k3，分别在第四序列和待对比第四序列中选择k3个最可靠位置对应的序号，分别基于选择的序号和已生成的第四中间结果序列获得编码码字性能结果，其中，第四中间结果序列中包含已选择的(k3-1)个序号，第三目标长度k3依次取值从t3到kmax^(m)，kmax^(m)为母码长度为m＝256时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新第四中间结果序列；将k3取值为kmax^(m)时获得的更新后的第四中间结果序列作为更新后第三子序列，将更新后第三子序列置于第四序列中，获得更新后第四序列。通俗的说，取q256^opt和q512，将q512中小于等于256的前(256-t3+1)个的元素集合记为p256-t3+1，在q256^opt中找到p256-t3+1所含元素的顺序，按照这个顺序替换第三子序列中前(256-t3+1)个序号在第三子序列中的顺序，得到长为512的序列q512*。t3为正整数，包括最小信息比特数加crc比特及pc比特数，例如，t3通过上述性能优劣判断规则给出的信息比特数和crc长度为例，t3＝8+19＝27。从k3＝t3开始，到256结束，每取一个k3的值，对比按照q512和q512*构造的polar码的性能，即(m＝512，k3)的polar码的性能，m为码长，k3为信息比特序号集合长度，可以理解，此时考察的(m＝512，k3)的polar码的性能也考虑到了(m＝256，k3)的polar码的性能。每选好一个性能较好的序号，依次放入q512^opt中，当q512和q512*两个序列构造的polar码的性能相同时，直接将这些序号放入q512^opt中。

最后，类似的，获取第五序列中序号小于等于512的第四子序列，按照第四子序列的前(512-t4+1)个序号在更新后第四序列中的排序，更新第四子序列的前(512-t4+1)个序号在第四子序列中的排序，基于更新后的第四子序列获得待对比第五序列；基于第四目标长度k4，分别在第五序列和待对比第五序列中选择k4个最可靠位置对应的序号，分别基于选择的序号和已生成的第五中间结果序列获得编码码字性能结果，其中，第五中间结果序列中包含已选择的(k4-1)个序号，第四目标长度k4依次取值从t4到kmax^(m)，kmax^(m)为母码长度为m＝512时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新第五中间结果序列；将k4取值为kmax^(m)时获得的更新后的第五中间结果序列作为更新后第四子序列，将更新后第四子序列置于第五序列中，获得更新后第五序列。通俗的说，取q512^opt和q1024，将q1024中小于等于512的前(512-t4+1)个的元素集合记为p512-t4+1，在q512^opt中找到p512-t4+1所含元素的顺序，按照这个顺序替换第四子序列中前(512-t4+1)个序号在第四子序列中的顺序，得到长为1024的序列q1024*。t4为正整数，包括最小信息比特数加crc比特及pc比特数，例如，t4通过上述性能优劣判断规则给出的信息比特数和crc长度为例，t4＝8+19＝27。从k4＝t4开始，到512结束，每取一个k4的值，对比按照q1024和q1024*构造的polar码的性能，即(m＝1024，k4)的polar码的性能，m为码长，k4为信息比特序号集合长度，可以理解，此时考察的(m＝1024，k4)的polar码的性能也考虑到了(m＝1024，k4)的polar码的性能。每选好一个性能较好的序号，依次放入q1024^opt中，当q1024和q1024*两个序列构造的polar码的性能相同时，直接将这些序号放入q1024^opt中。

更新后第五序列即为本申请实施例中用于对待编码比特进行编码的排序序列。该排序序列对应最大母码长度为1024。可选的，由于上述方法获得的排序序列符合嵌套特性，因此当最大母码长度的排序序列确定，可以根据该排序序列获得其它较短母码长度的排序序列，例如，获得母码长度为64、128、256、512的排序序列。

下面具体介绍一下如何获得上述步骤一种的第一序列～第五序列。

具体地，根据第五目标长度k5，在(m-k5+1)个序号中依次选择一个序号，每选择一个序号，基于所选择的一个序号与已生成的中间结果序列获得编码码字性能结果，其中，中间结果序列包含已选择的(k5-1)个序号，(m-k5+1)个序号与中间结果序列不相交，目标长度k5依次取值从1到kmax^(m)，kmax^(m)为母码长度为m时的最大信息比特长度，kmax为系统支持的最大信息比特向量长度，例如，kmax为上述性能优劣判断规则中所述的值为5n/6-(j+j′)，这里的n实际可以认为为m，即不同码长下kmax取值不同；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新中间结果序列。m为2的整数次幂，m为母码长度，母码长度分别为64、128、256、512、1024。

例如，m＝64时，中间结果序列qn的初始值置为空集，k5的初始值置为0，执行k5＝k5+1，选择极化信道序号ik5，其中，ik5∈[1，n]且也就是，在未确定的极化信道中选则一个极化信道，将所选的ik5置于序列qn的首位，使得更新后的qn对于(m＝64，k＝k5)的polar码的性能是按照上述性能优劣判断规则得到的最优序列，判断k5是否大于等于kmax，如果不是，则继续返回执行k5＝k5+1，若是，则结束操作。

又例如，m＝128时，中间结果序列qn的初始值置为空集，k5的初始值置为0，执行k5＝k5+1，选择极化信道序号ik5，其中，ik5∈[1，n]且也就是，在未确定的极化信道中选则一个极化信道，将所选的ik5置于序列qn的首位，使得更新后的qn对于(m＝128，k＝k5)的polar码的性能是按照上述性能优劣判断规则得到的最优序列，判断k5是否大于等于kmax，如果不是，则继续返回执行k5＝k5+1，若是，则结束操作。

如此，可以得到母码长度分别为64、128、256、512、1024的序列，即可得到上述第一序列～第五序列。

如上所述，本申请实施例在得到q序列后，可以根据q序列获得z序列。

本申请实施例提供如下一些可选的排序序列的举例。使用如下排序序列能够有助于提高polar码的编译性能，如下序列可以采用本申请实施例提供的方法获得，也可以通过其他可能的方法获得，本申请实施例不作限定。

n＝1024，排序序列为q序列，记为序列①:

[1,2,3,5,9,17,33,4,6,65,7,10,11,129,18,13,19,34,257,21,35,25,66,37,67,513,8,12,130,41,69,14,20,131,49,15,73,258,22,133,36,81,23,259,26,137,38,261,97,145,27,514,39,515,265,68,42,29,161,70,43,517,50,71,273,193,521,529,45,16,132,74,51,289,545,24,75,53,134,82,321,77,135,28,57,83,138,260,40,98,385,30,85,139,262,146,31,99,516,44,89,141,263,147,101,266,72,46,162,518,577,519,52,149,522,47,76,267,163,274,523,641,105,54,153,78,269,194,165,275,55,84,530,525,531,136,58,113,195,290,79,169,277,59,86,61,87,140,100,197,90,291,281,546,769,142,177,91,102,148,143,533,547,537,32,103,264,48,322,293,201,323,578,150,297,93,106,209,549,579,268,386,325,305,164,151,107,56,154,225,387,329,270,114,520,553,642,80,166,109,155,276,60,271,115,167,524,581,561,157,196,170,198,171,278,292,279,117,88,62,178,282,526,643,532,527,63,92,585,770,144,389,294,199,104,337,173,283,202,121,179,295,94,534,645,535,593,548,771,152,324,393,298,203,108,285,181,210,538,95,205,299,353,649,609,326,185,211,401,306,301,110,156,327,116,111,773,550,657,539,551,168,213,307,330,158,226,118,172,331,227,388,309,217,417,338,159,777,272,580,119,541,554,280,333,390,174,122,200,180,229,339,313,391,64,394,284,175,233,341,449,354,395,204,123,673,555,557,562,582,296,286,182,206,125,785,705,183,287,528,583,644,586,563,300,355,212,402,186,397,345,207,302,801,96,187,241,536,587,565,646,594,328,403,214,357,308,303,833,589,647,112,540,569,595,650,772,228,418,332,215,405,310,189,552,610,897,160,218,361,419,409,369,230,450,421,542,597,651,774,658,311,120,334,219,543,611,176,392,231,314,124,340,335,221,451,425,315,556,601,653,234,775,659,613,235,288,453,126,342,396,778,584,558,674,356,343,564,661,559,617,779,675,184,398,433,317,346,242,208,404,358,786,188,588,566,665,625,781,237,127,243,347,399,706,216,677,787,590,567,648,406,359,304,570,596,362,707,190,457,349,465,802,420,407,245,410,591,681,789,571,598,363,573,220,422,599,652,612,709,602,803,370,793,312,191,654,689,603,411,232,249,365,336,423,614,660,655,316,222,371,426,452,413,481,236,223,344,318,544,373,615,427,776,128,454,238,560,834,805,713,835,662,809,605,618,721,780,434,898,319,455,837,817,676,663,348,429,400,377,244,239,458,435,360,568,619,666,737,899,841,782,626,350,246,459,592,678,621,667,788,572,783,627,679,408,364,351,192,247,437,466,412,461,250,366,600,708,574,669,682,790,804,791,710,683,629,441,690,794,604,575,467,424,372,251,414,367,469,482,374,656,901,806,711,428,415,253,616,849,685,795,714,633,691,807,606,483,473,224,664,836,905,810,715,620,797,693,375,485,430,456,722,607,717,811,865,838,697,723,913,818,378,436,813,320,622,240,431,460,379,668,839,489,438,628,623,462,381,439,352,497,680,725,819,842,670,738,630,468,248,442,463,470,443,252,684,843,739,900,821,729,929,850,671,784,631,792,845,902,686,368,634,712,254,692,741,851,825,903,687,471,416,484,445,376,474,906,796,486,635,745,853,961,866,907,716,694,808,475,798,637,695,255,718,576,799,812,867,432,380,698,914,487,724,909,857,719,477,814,608,490,699,753,840,915,726,869,820,815,440,491,624,382,498,493,464,672,930,844,740,917,822,727,632,701,873,931,921,881,730,383,499,444,446,472,962,933,823,742,846,731,447,688,904,636,826,743,852,847,256,476,733,963,937,827,746,501,488,638,505,800,696,854,908,868,855,747,910,829,858,754,720,916,870,478,700,749,639,816,965,945,755,859,911,492,479,728,918,871,384,494,500,874,702,969,495,861,502,932,919,757,922,875,732,977,934,882,761,824,703,923,744,877,848,935,828,734,883,448,938,503,964,925,993,748,735,856,885,939,506,966,507,830,750,946,860,831,967,756,941,912,872,751,889,480,970,947,862,758,971,509,920,640,876,863,759,949,978,924,973,762,878,979,496,936,704,884,953,763,504,926,879,981,994,886,940,927,765,736,887,995,942,968,985,508,890,948,832,752,943,997,972,891,1001,510,950,974,893,951,864,760,1009,511,980,954,764,975,955,880,982,983,928,996,766,957,888,986,998,987,944,892,999,767,512,989,1002,952,1003,894,976,895,1010,956,1005,1011,958,984,959,988,1013,1000,1017,768,990,1004,991,1006,1012,1007,1014,896,960,1015,1018,1019,992,1021,1008,1016,1020,1022,1023,1024]。

n＝1024，排序序列为z序列，记为序列②:

[1,2,3,8,4,9,11,27,5,12,13,28,16,32,36,70,6,15,17,33,20,39,43,76,22,45,51,84,58,92,97,165,7,18,21,41,24,47,53,89,30,57,61,100,69,108,116,168,35,63,73,113,78,124,131,188,85,137,144,203,146,220,227,315,10,23,25,56,31,60,64,107,37,72,77,117,82,126,141,198,42,80,86,132,93,145,147,219,101,151,158,228,176,243,260,355,49,90,98,149,105,159,166,235,123,177,187,255,200,272,276,372,138,194,205,275,218,288,301,403,240,308,325,412,335,429,461,535,14,29,34,71,40,79,83,136,46,87,94,148,102,156,161,231,50,96,104,160,114,174,186,250,125,189,201,273,210,286,297,388,59,109,119,185,129,199,206,282,142,212,214,289,237,307,318,408,157,221,241,310,257,333,338,444,267,349,356,454,384,479,506,589,66,128,139,211,150,213,234,309,171,239,254,324,261,334,352,450,178,258,268,347,283,365,381,466,294,389,405,495,415,520,527,640,190,287,291,378,311,394,410,511,319,422,426,526,460,537,562,668,357,449,462,561,486,575,590,691,512,595,616,696,628,715,744,811,19,38,44,88,48,95,103,167,55,106,118,181,127,193,204,299,65,120,130,202,143,215,217,304,153,222,238,317,256,332,339,427,74,140,152,216,170,233,242,331,175,253,262,345,271,353,368,474,184,270,284,367,293,383,402,505,313,411,418,519,447,529,551,666,81,169,172,251,183,266,274,363,192,285,290,380,305,404,414,514,236,296,312,413,320,430,437,528,351,448,463,557,481,574,588,681,263,322,346,436,366,452,473,565,390,477,493,587,513,596,618,712,393,503,521,615,531,621,649,726,560,663,671,751,679,774,792,851,91,182,191,292,232,306,314,409,252,316,323,431,350,445,464,559,269,348,364,451,382,472,485,586,392,487,510,593,524,617,627,723,295,379,391,484,396,496,515,614,417,522,533,626,558,651,669,750,446,549,564,664,591,675,680,771,608,692,695,794,725,795,803,880,321,395,416,523,428,536,552,652,480,563,576,670,594,678,693,777,482,592,613,690,619,694,722,796,639,727,740,812,759,836,847,906,525,620,638,724,650,730,754,819,674,762,772,846,776,852,857,925,682,775,793,853,818,859,882,931,821,891,893,946,912,956,964,985,26,52,54,99,62,110,112,195,67,115,121,207,134,223,226,340,68,133,135,225,162,244,246,358,164,259,280,373,302,397,406,530,75,154,163,248,179,278,281,385,196,303,327,419,328,434,440,538,209,329,344,438,360,456,470,566,374,475,491,582,494,599,612,746,111,173,180,300,208,330,341,433,229,343,359,455,370,469,488,577,247,362,375,476,398,492,497,597,420,501,509,611,545,637,654,761,265,386,407,499,425,516,532,629,441,546,567,646,579,667,677,773,458,573,584,676,607,689,707,785,634,713,731,806,742,820,839,914,122,197,224,342,245,361,371,471,264,376,399,498,421,507,518,622,279,401,424,517,439,543,556,641,457,568,580,672,600,687,705,778,326,435,443,555,467,578,585,683,489,601,606,697,631,711,721,804,508,609,635,716,648,738,743,823,659,752,763,837,786,855,871,927,337,465,478,598,500,605,625,714,541,633,645,737,655,745,758,833,547,653,660,755,684,767,784,848,702,791,802,865,813,878,887,940,569,688,699,781,717,800,808,873,732,817,828,886,838,895,904,950,764,832,843,900,862,910,917,962,869,922,930,967,939,975,984,1004,155,230,249,377,277,400,423,534,298,432,442,548,459,572,583,706,336,453,468,581,490,602,604,708,504,610,632,729,647,741,747,822,354,483,502,603,540,624,636,739,544,644,656,748,665,760,770,840,554,662,685,769,701,783,799,870,719,807,816,877,830,894,898,949,369,539,542,642,553,658,673,765,571,686,698,780,709,801,810,875,630,704,718,809,733,824,827,888,757,831,844,897,858,909,916,961,657,735,749,826,768,835,850,903,787,854,864,915,874,923,933,970,790,868,879,928,889,936,941,977,905,947,954,982,959,990,992,1012,387,550,570,700,623,710,720,805,643,728,736,825,756,829,845,902,661,753,766,834,782,849,861,913,789,863,872,920,884,932,938,973,703,779,788,860,798,867,876,926,815,881,890,937,901,943,951,981,842,896,908,948,918,957,960,988,929,966,969,994,976,997,999,1013,734,797,814,883,841,892,899,944,856,907,911,953,921,958,968,991,866,919,924,965,934,971,972,998,945,978,980,1000,986,1005,1007,1017,885,935,942,974,952,979,983,1002,955,987,989,1006,995,1008,1010,1019,963,993,996,1009,1001,1011,1014,1020,1003,1015,1016,1021,1018,1022,1023,1024]。

n＝1024，排序序列为q序列，记为序列③:

[1,2,3,5,9,17,33,4,6,65,7,10,11,129,18,13,19,34,257,21,35,25,66,37,67,513,8,12,130,41,69,14,20,131,49,15,73,258,22,133,36,81,23,259,26,137,38,261,97,145,27,514,39,515,265,68,42,29,161,70,43,517,50,71,273,45,521,529,16,132,74,193,51,289,545,24,75,53,134,82,321,77,135,28,57,83,138,260,40,98,385,85,30,139,262,146,31,99,516,44,89,141,263,147,101,266,72,46,162,518,577,519,52,149,522,47,76,267,163,274,523,641,105,54,153,78,269,194,165,275,55,84,530,525,531,136,58,113,195,290,79,169,277,59,86,100,87,140,61,197,90,291,281,546,769,142,177,91,102,148,143,533,547,537,32,103,264,106,322,293,201,323,578,150,297,48,93,209,549,579,268,386,325,305,164,151,107,56,154,225,387,329,270,114,520,553,642,80,166,109,155,276,60,271,115,167,524,581,561,157,196,170,198,171,278,292,279,117,88,62,178,282,526,643,532,527,63,92,585,770,144,389,294,199,104,337,173,283,202,121,179,295,94,534,645,535,593,548,771,152,324,393,298,203,108,285,181,210,538,95,205,299,353,649,609,326,185,211,401,306,301,110,156,327,116,111,773,550,657,539,551,168,213,307,330,158,226,118,172,331,227,388,309,217,417,338,159,777,272,580,119,541,554,280,333,390,174,122,200,180,229,339,313,391,64,394,284,175,123,341,449,354,395,233,204,673,555,557,562,582,296,286,182,206,125,785,705,183,287,528,583,644,586,563,300,355,212,402,186,397,345,207,302,801,96,187,241,536,587,565,646,594,328,403,214,357,308,303,833,589,647,112,540,569,595,650,772,228,418,332,215,405,310,189,552,610,897,160,218,361,419,409,369,230,450,421,542,597,651,774,658,311,120,334,219,543,611,231,392,234,314,176,340,335,221,451,425,315,556,601,653,124,775,659,613,235,288,453,126,342,396,778,584,558,674,356,343,564,661,559,617,779,675,184,398,433,317,346,242,208,404,358,786,188,588,566,665,625,781,237,127,243,347,399,706,216,677,787,590,567,648,406,359,304,570,596,362,707,190,457,349,465,802,420,407,245,410,591,681,789,571,598,363,573,220,422,599,652,612,709,602,803,370,793,312,191,654,689,603,411,232,249,365,336,423,614,660,655,316,222,371,426,452,413,481,236,223,344,318,544,373,615,427,776,128,454,238,560,834,805,713,835,662,809,605,618,721,780,434,898,319,455,837,817,676,663,348,429,400,377,244,239,458,435,360,568,619,666,737,899,841,782,626,350,246,459,592,678,621,667,788,572,783,627,679,408,364,351,192,247,437,466,412,461,250,366,600,708,574,669,682,790,804,791,710,683,629,441,690,794,604,575,467,424,372,251,482,414,469,367,374,656,901,806,711,428,415,253,616,849,685,795,714,633,691,807,606,483,473,224,664,836,905,810,715,620,797,693,375,485,430,456,722,607,717,811,865,838,697,723,913,818,378,436,813,320,622,240,431,460,379,668,839,489,438,628,623,462,381,439,352,497,680,725,819,842,670,738,630,468,248,442,463,470,443,252,684,843,739,900,821,729,929,850,671,784,631,792,845,902,686,368,634,712,254,692,741,851,825,903,687,471,416,484,445,376,474,906,796,486,635,745,853,961,866,907,716,694,808,475,798,637,695,255,718,576,799,812,867,432,380,698,914,487,724,909,857,719,477,814,608,490,699,753,840,915,726,869,820,815,440,491,624,382,498,493,464,672,930,844,740,917,822,727,632,701,873,931,921,881,730,383,499,444,446,472,962,933,823,742,846,731,447,688,904,636,826,743,852,847,256,476,733,963,937,827,746,501,488,638,505,800,696,854,908,868,855,747,910,829,858,754,720,916,870,478,700,749,639,816,965,945,755,859,911,492,479,728,918,871,384,494,500,874,702,969,495,861,502,932,919,757,922,875,732,977,934,882,761,824,703,923,744,877,848,935,828,734,883,448,938,503,964,925,993,748,735,856,885,939,506,966,507,830,750,946,860,831,967,756,941,912,872,751,889,480,970,947,862,758,971,509,920,640,876,863,759,949,978,924,973,762,878,979,496,936,704,884,953,763,504,926,879,981,994,886,940,927,765,736,887,995,942,968,985,508,890,948,832,752,943,997,972,891,1001,510,950,974,893,951,864,760,1009,511,980,954,764,975,955,880,982,983,928,996,766,957,888,986,998,987,944,892,999,767,512,989,1002,952,1003,894,976,895,1010,956,1005,1011,958,984,959,988,1013,1000,1017,768,990,1004,991,1006,1012,1007,1014,896,960,1015,1018,1019,992,1021,1008,1016,1020,1022,1023,1024]。

n＝1024，排序序列为z序列，记为序列④:

[1,2,3,8,4,9,11,27,5,12,13,28,16,32,36,69,6,15,17,33,20,39,43,76,22,45,51,84,58,93,97,165,7,18,21,41,24,47,53,89,30,57,61,100,66,108,116,176,35,63,73,113,78,124,131,188,85,137,144,203,149,220,227,315,10,23,25,56,31,60,64,107,37,71,77,117,82,126,141,198,42,80,86,132,92,145,147,219,101,151,158,228,177,243,260,355,49,90,98,146,105,159,166,235,123,168,187,255,200,272,276,372,138,194,205,275,218,288,301,403,240,308,319,422,335,429,461,535,14,29,34,70,40,79,83,136,46,87,94,148,102,156,161,231,50,96,104,160,114,174,186,250,125,189,201,273,210,286,297,388,59,109,119,185,129,199,206,282,142,212,214,289,237,307,318,412,157,221,241,310,257,333,338,444,267,349,356,454,384,479,506,589,72,128,139,211,150,213,234,309,171,239,254,325,261,334,352,450,178,258,268,347,283,365,381,466,294,389,405,495,415,520,527,640,190,287,291,378,311,394,408,511,324,410,426,526,460,537,562,668,357,449,462,561,486,575,590,691,512,595,616,696,628,715,744,811,19,38,44,88,48,95,103,167,55,106,118,181,127,193,204,299,65,120,130,202,143,215,217,304,153,222,238,317,256,332,339,427,74,140,152,216,170,233,242,331,175,253,262,345,271,353,368,474,184,270,284,367,293,383,402,505,313,411,418,519,447,529,551,666,81,169,172,251,183,266,274,363,192,285,290,380,305,404,414,514,236,296,312,413,320,430,437,528,351,448,463,557,481,574,588,681,263,322,346,436,366,452,473,565,390,477,493,587,513,596,620,712,393,503,521,615,531,621,649,726,560,663,671,751,679,774,792,851,91,182,191,292,232,306,314,409,252,316,323,431,350,445,464,559,269,348,364,451,382,472,485,586,392,487,510,593,524,618,627,723,295,379,391,484,396,496,515,614,417,522,533,626,558,651,669,750,446,549,564,664,591,675,680,771,608,692,695,794,725,795,803,880,321,395,416,523,428,536,552,652,480,563,576,670,594,678,693,777,482,592,613,690,619,694,722,796,639,727,740,812,759,836,847,906,525,617,638,724,650,730,754,819,674,762,772,846,776,852,857,925,682,775,793,853,818,859,882,931,821,891,893,946,912,956,964,985,26,52,54,99,62,110,112,195,67,115,121,207,134,223,226,340,68,133,135,225,162,244,246,358,164,259,280,373,302,397,406,530,75,154,163,248,179,278,281,385,196,303,327,419,328,434,440,538,209,329,344,438,360,456,470,566,374,475,491,582,494,599,612,746,111,173,180,300,208,330,341,433,229,343,359,455,370,469,488,577,247,362,375,476,398,492,497,597,420,501,509,611,545,637,654,761,265,386,407,499,425,516,532,629,441,546,567,646,579,667,677,773,458,573,584,676,607,689,707,785,634,713,731,806,742,820,839,914,122,197,224,342,245,361,371,471,264,376,399,498,421,507,518,622,279,401,424,517,439,543,556,641,457,568,580,672,600,687,705,778,326,435,443,555,467,578,585,683,489,601,606,697,631,711,721,804,508,609,635,716,648,738,743,823,659,752,763,837,786,855,871,927,337,465,478,598,500,605,625,714,541,633,645,737,655,745,758,833,547,653,660,755,684,767,784,848,702,791,802,865,813,878,887,940,569,688,699,781,717,800,808,873,732,817,828,886,838,895,904,950,764,832,843,900,862,910,917,962,869,922,930,967,939,975,984,1004,155,230,249,377,277,400,423,534,298,432,442,548,459,572,583,706,336,453,468,581,490,602,604,708,504,610,632,729,647,741,747,822,354,483,502,603,540,624,636,739,544,644,656,748,665,760,770,840,554,662,685,769,701,783,799,870,719,807,816,877,830,894,898,949,369,539,542,642,553,658,673,765,571,686,698,780,709,801,810,875,630,704,718,809,733,824,827,888,757,831,844,897,858,909,916,961,657,735,749,826,768,835,850,903,787,854,864,915,874,923,933,970,790,868,879,928,889,936,941,977,905,947,954,982,959,990,992,1012,387,550,570,700,623,710,720,805,643,728,736,825,756,829,845,902,661,753,766,834,782,849,861,913,789,863,872,920,884,932,938,973,703,779,788,860,798,867,876,926,815,881,890,937,901,943,951,981,842,896,908,948,918,957,960,988,929,966,969,994,976,997,999,1013,734,797,814,883,841,892,899,944,856,907,911,953,921,958,968,991,866,919,924,965,934,971,972,998,945,978,980,1000,986,1005,1007,1017,885,935,942,974,952,979,983,1002,955,987,989,1006,995,1008,1010,1019,963,993,996,1009,1001,1011,1014,1020,1003,1015,1016,1021,1018,1022,1023,1024]。

n＝1024，排序序列为q序列，记为序列⑤:

[1,2,5,9,3,17,33,7,65,513,4,6,10,13,129,19,34,18,11,21,35,25,257,66,37,8,67,130,12,41,14,133,514,20,69,49,15,258,73,22,131,27,81,137,259,36,39,23,38,517,97,26,68,265,42,145,29,70,261,50,43,161,75,521,71,45,82,132,273,193,16,51,74,24,53,515,135,138,289,77,134,83,28,40,260,98,529,57,139,85,30,146,262,44,321,545,99,141,266,31,89,147,263,72,519,162,46,101,385,149,52,47,577,76,267,54,274,165,194,105,516,163,269,78,153,275,55,525,58,113,136,79,86,290,518,195,84,291,169,277,59,531,140,100,323,87,61,641,197,90,177,281,148,102,522,293,142,322,143,91,546,201,32,103,264,530,106,387,48,297,523,93,150,209,154,268,164,537,769,533,325,114,151,166,107,305,56,578,276,549,270,386,155,80,109,225,167,60,579,170,115,196,329,271,278,157,547,117,88,389,520,337,292,279,198,642,62,178,171,553,199,282,144,63,524,92,173,295,585,202,393,104,645,121,294,283,532,353,179,203,561,324,298,94,581,210,181,152,526,285,108,401,770,95,205,299,527,327,156,534,306,110,326,643,211,116,185,539,111,301,593,168,388,330,548,226,417,771,158,272,213,551,307,118,649,172,227,331,609,338,390,535,309,217,550,159,391,538,119,280,333,580,174,122,777,339,180,554,200,657,354,284,541,313,449,582,229,394,773,64,175,673,555,785,395,562,204,182,341,583,557,123,296,286,233,125,183,212,287,300,355,206,402,186,397,587,345,594,647,528,96,563,241,644,207,705,328,586,403,536,214,357,308,419,565,801,589,187,302,646,112,569,303,595,361,228,650,772,418,540,215,405,310,189,450,332,218,610,597,552,651,160,230,120,334,409,311,421,219,542,369,658,340,774,231,392,556,176,833,314,221,611,653,335,451,778,124,543,342,601,425,315,659,234,775,184,613,356,674,126,396,584,235,558,317,564,661,242,779,559,343,346,617,675,897,404,288,398,188,208,453,433,786,237,588,566,665,781,347,362,127,243,358,706,406,677,590,216,399,567,304,598,359,802,420,625,457,787,349,190,570,245,591,411,648,220,707,312,596,363,803,465,681,407,789,422,599,232,571,370,652,249,835,410,709,191,614,316,573,365,660,423,336,222,689,452,793,371,612,426,602,236,805,413,654,481,834,223,713,344,603,318,544,427,655,454,615,373,776,434,560,238,899,618,348,809,244,721,455,666,319,605,377,662,429,780,239,676,360,837,400,626,246,663,678,435,568,458,817,350,619,459,788,408,782,898,128,667,364,592,466,621,437,737,247,679,572,250,682,351,627,461,708,841,412,783,366,790,441,600,424,669,629,467,901,251,849,804,372,791,192,710,482,574,690,414,683,367,604,794,469,714,374,711,428,575,656,253,807,415,685,905,483,616,224,633,806,375,795,865,430,691,456,715,836,473,810,378,606,620,436,664,722,320,797,431,693,913,240,607,717,485,811,379,839,668,438,818,622,460,838,723,248,697,381,738,680,462,813,628,489,900,842,442,623,929,352,725,784,470,630,819,439,670,463,739,684,252,843,850,497,902,821,729,468,634,903,368,671,792,443,845,631,475,686,851,484,692,712,380,866,796,416,825,961,741,254,906,635,445,694,745,486,808,687,907,471,576,716,376,867,914,474,853,637,798,432,695,812,487,753,724,799,490,857,909,255,718,608,931,477,698,726,915,440,820,840,869,493,719,699,382,814,624,815,499,873,740,930,672,917,822,446,727,962,844,464,632,730,701,491,742,846,921,383,823,852,731,472,881,743,498,904,826,444,688,476,636,933,847,963,733,937,501,854,827,447,746,638,868,800,908,488,696,747,829,858,754,855,916,494,910,720,478,749,965,945,639,870,911,700,859,505,755,728,918,871,479,500,874,932,969,702,816,861,256,757,492,875,824,732,923,977,919,934,744,761,882,503,703,922,828,877,935,848,734,964,883,495,938,748,506,925,993,856,735,384,502,966,939,885,830,750,946,860,756,480,967,831,889,941,751,872,507,971,912,758,947,970,862,978,448,876,920,640,759,949,863,762,509,973,924,878,953,887,936,979,763,504,884,704,994,926,879,981,942,765,496,927,886,995,736,940,985,968,890,948,832,508,943,752,974,997,891,950,760,893,972,1001,954,510,864,982,951,975,764,1009,980,880,955,987,996,892,928,511,766,957,998,983,888,986,944,999,1002,767,989,952,1005,894,1011,958,976,512,1003,895,984,1010,956,988,1013,959,1000,1006,990,1017,991,1012,768,1004,1015,1007,1018,896,1014,992,1019,960,1021,1016,1008,1020,1022,1023,1024]。

n＝1024，排序序列为z序列，记为序列⑥:

[1,2,5,9,3,17,33,7,65,513,4,6,10,13,129,19,34,18,11,21,35,25,257,66,37,8,67,130,12,41,14,133,514,20,69,49,15,258,73,22,131,27,81,137,259,36,39,23,38,517,97,26,68,265,42,145,29,70,261,50,43,161,75,521,71,45,82,132,273,193,16,51,74,24,53,515,135,138,289,77,134,83,28,40,260,98,529,57,139,85,30,146,262,44,321,545,99,141,266,31,89,147,263,72,519,162,46,101,385,149,52,47,577,76,267,54,274,165,194,105,516,163,269,78,153,275,55,525,58,113,136,79,86,290,518,195,84,291,169,277,59,531,140,100,323,87,61,641,197,90,177,281,148,102,522,293,142,322,143,91,546,201,32,103,264,530,106,387,48,297,523,93,150,209,154,268,164,537,769,533,325,114,151,166,107,305,56,578,276,549,270,386,155,80,109,225,167,60,579,170,115,196,329,271,278,157,547,117,88,389,520,337,292,279,198,642,62,178,171,553,199,282,144,63,524,92,173,295,585,202,393,104,645,121,294,283,532,353,179,203,561,324,298,94,581,210,181,152,526,285,108,401,770,95,205,299,527,327,156,534,306,110,326,643,211,116,185,539,111,301,593,168,388,330,548,226,417,771,158,272,213,551,307,118,649,172,227,331,609,338,390,535,309,217,550,159,391,538,119,280,333,580,174,122,777,339,180,554,200,657,354,284,541,313,449,582,229,394,773,64,175,673,555,785,395,562,204,182,341,583,557,123,296,286,233,125,183,212,287,300,355,206,402,186,397,587,345,594,647,528,96,563,241,644,207,705,328,586,403,536,214,357,308,419,565,801,589,187,302,646,112,569,303,595,361,228,650,772,418,540,215,405,310,189,450,332,218,610,597,552,651,160,230,120,334,409,311,421,219,542,369,658,340,774,231,392,556,176,833,314,221,611,653,335,451,778,124,543,342,601,425,315,659,234,775,184,613,356,674,126,396,584,235,558,317,564,661,242,779,559,343,346,617,675,897,404,288,398,188,208,453,433,786,237,588,566,665,781,347,362,127,243,358,706,406,677,590,216,399,567,304,598,359,802,420,625,457,787,349,190,570,245,591,411,648,220,707,312,596,363,803,465,681,407,789,422,599,232,571,370,652,249,835,410,709,191,614,316,573,365,660,423,336,222,689,452,793,371,612,426,602,236,805,413,654,481,834,223,713,344,603,318,544,427,655,454,615,373,776,434,560,238,899,618,348,809,244,721,455,666,319,605,377,662,429,780,239,676,360,837,400,626,246,663,678,435,568,458,817,350,619,459,788,408,782,898,128,667,364,592,466,621,437,737,247,679,572,250,682,351,627,461,708,841,412,783,366,790,441,600,424,669,629,467,901,251,849,804,372,791,192,710,482,574,690,414,683,367,604,794,469,714,374,711,428,575,656,253,807,415,685,905,483,616,224,633,806,375,795,865,430,691,456,715,836,473,810,378,606,620,436,664,722,320,797,431,693,913,240,607,717,485,811,379,839,668,438,818,622,460,838,723,248,697,381,738,680,462,813,628,489,900,842,442,623,929,352,725,784,470,630,819,439,670,463,739,684,252,843,850,497,902,821,729,468,634,903,368,671,792,443,845,631,475,686,851,484,692,712,380,866,796,416,825,961,741,254,906,635,445,694,745,486,808,687,907,471,576,716,376,867,914,474,853,637,798,432,695,812,487,753,724,799,490,857,909,255,718,608,931,477,698,726,915,440,820,840,869,493,719,699,382,814,624,815,499,873,740,930,672,917,822,446,727,962,844,464,632,730,701,491,742,846,921,383,823,852,731,472,881,743,498,904,826,444,688,476,636,933,847,963,733,937,501,854,827,447,746,638,868,800,908,488,696,747,829,858,754,855,916,494,910,720,478,749,965,945,639,870,911,700,859,505,755,728,918,871,479,500,874,932,969,702,816,861,256,757,492,875,824,732,923,977,919,934,744,761,882,503,703,922,828,877,935,848,734,964,883,495,938,748,506,925,993,856,735,384,502,966,939,885,830,750,946,860,756,480,967,831,889,941,751,872,507,971,912,758,947,970,862,978,448,876,920,640,759,949,863,762,509,973,924,878,953,887,936,979,763,504,884,704,994,926,879,981,942,765,496,927,886,995,736,940,985,968,890,948,832,508,943,752,974,997,891,950,760,893,972,1001,954,510,864,982,951,975,764,1009,980,880,955,987,996,892,928,511,766,957,998,983,888,986,944,999,1002,767,989,952,1005,894,1011,958,976,512,1003,895,984,1010,956,988,1013,959,1000,1006,990,1017,991,1012,768,1004,1015,1007,1018,896,1014,992,1019,960,1021,1016,1008,1020,1022,1023,1024]。

n＝1024，排序序列为q序列，记为序列⑦:

[1,2,5,9,3,17,33,7,65,513,4,13,6,19,129,10,34,18,11,25,21,35,257,66,37,8,67,130,12,41,14,133,514,20,69,49,15,258,73,22,131,81,27,36,259,137,39,97,23,517,38,26,68,265,42,145,29,70,261,43,50,161,75,521,71,45,82,132,273,193,16,51,74,24,53,515,135,138,289,77,134,83,28,40,260,98,529,57,139,85,30,146,262,44,321,545,99,141,266,31,89,147,263,72,519,162,46,101,385,149,52,47,577,76,267,54,274,165,194,105,516,163,269,78,153,275,55,525,58,113,136,79,86,290,518,195,84,291,59,277,169,531,100,140,323,197,87,641,177,61,90,281,102,148,522,293,142,322,143,91,546,201,32,103,264,530,106,387,209,297,523,48,154,93,150,268,164,537,769,533,325,114,151,166,107,305,56,578,276,549,270,386,155,80,225,167,109,115,579,170,60,196,329,271,278,157,547,88,117,389,520,337,292,279,198,642,62,178,171,553,199,282,144,63,524,92,173,295,585,202,393,104,645,121,294,283,532,353,179,203,561,324,298,94,581,152,210,181,526,285,108,401,770,95,205,299,527,327,156,534,306,110,326,643,211,116,185,539,168,301,593,226,388,330,548,111,417,771,158,272,213,551,307,118,649,172,227,331,609,338,390,535,309,217,550,159,391,538,119,280,333,580,174,122,777,339,180,554,200,657,354,284,541,313,449,582,229,394,773,123,175,673,555,785,395,562,204,182,341,583,557,64,296,286,233,125,183,212,287,300,355,206,402,186,397,587,345,594,647,528,96,563,241,644,207,705,328,586,403,536,214,357,308,419,565,801,589,187,302,646,112,569,303,595,361,228,650,772,418,540,215,405,310,189,450,332,218,610,597,552,651,160,230,120,334,409,311,421,219,542,369,658,340,774,231,392,556,176,833,314,221,611,653,335,451,778,124,543,342,601,425,315,659,234,775,184,613,356,674,126,396,584,235,558,317,564,661,242,779,559,343,346,617,675,897,404,288,398,188,453,433,208,786,237,588,566,665,781,347,362,127,243,358,706,406,677,590,216,399,567,304,598,359,802,420,625,457,787,349,245,570,190,591,411,648,220,707,312,596,363,803,465,681,407,789,422,599,249,571,370,652,191,835,410,709,232,614,316,573,365,660,423,336,222,689,452,793,371,612,426,602,236,805,413,654,481,834,223,713,344,603,454,544,318,655,427,615,373,776,434,560,238,899,618,348,809,244,721,455,666,319,605,377,662,429,780,239,676,360,837,400,626,246,663,678,435,568,458,817,350,619,459,788,408,782,898,128,667,364,592,466,621,437,737,247,679,572,250,682,351,627,461,708,841,412,783,366,790,441,600,424,669,629,467,901,251,849,804,372,791,192,710,482,574,690,414,683,367,604,794,469,714,374,711,253,575,656,415,807,483,685,905,375,616,224,633,806,428,795,865,430,691,456,715,836,473,810,378,606,620,436,664,722,320,797,431,693,913,240,607,717,485,811,379,839,668,438,818,622,460,838,723,248,697,381,738,680,462,813,628,489,900,842,442,623,929,352,725,784,470,630,819,439,670,463,739,684,252,843,850,497,902,821,729,468,634,903,368,671,792,443,845,631,475,686,851,484,692,712,380,866,796,416,825,961,741,254,906,635,445,694,745,486,808,687,907,471,576,716,376,867,914,474,853,637,798,432,695,812,487,753,724,799,490,857,909,255,718,608,931,477,698,726,915,440,820,840,869,493,719,699,382,814,624,815,499,873,740,930,672,917,822,464,727,962,844,491,632,730,701,383,742,846,921,472,823,852,731,498,881,743,444,904,826,501,688,446,636,933,847,963,733,937,476,854,827,447,746,868,638,800,908,488,696,747,829,858,754,855,916,965,478,910,720,505,749,945,639,870,911,700,859,479,755,918,871,256,728,494,874,932,969,702,816,861,500,757,923,492,732,824,875,977,919,934,744,761,882,495,703,922,828,877,935,848,503,734,883,938,384,964,748,925,993,506,856,735,502,830,966,939,885,946,750,860,756,480,967,831,889,941,751,872,507,971,912,758,947,970,862,978,448,876,920,640,759,949,863,762,509,973,924,878,953,887,936,979,763,504,884,704,994,926,879,981,942,765,496,927,886,995,736,940,985,968,890,948,832,508,943,752,974,997,891,950,760,893,972,1001,954,510,864,982,951,975,764,1009,980,880,955,987,996,892,928,511,766,957,998,983,888,986,944,999,1002,767,989,952,1005,894,1011,958,976,512,1003,895,984,1010,956,988,1013,959,1000,1006,990,1017,991,1012,768,1004,1015,1007,1018,896,1014,992,1019,960,1021,1016,1008,1020,1022,1023,1024]。

n＝1024，排序序列为z序列，记为序列⑧:

[1,2,5,9,3,17,33,7,65,513,4,13,6,19,129,10,34,18,11,25,21,35,257,66,37,8,67,130,12,41,14,133,514,20,69,49,15,258,73,22,131,81,27,36,259,137,39,97,23,517,38,26,68,265,42,145,29,70,261,43,50,161,75,521,71,45,82,132,273,193,16,51,74,24,53,515,135,138,289,77,134,83,28,40,260,98,529,57,139,85,30,146,262,44,321,545,99,141,266,31,89,147,263,72,519,162,46,101,385,149,52,47,577,76,267,54,274,165,194,105,516,163,269,78,153,275,55,525,58,113,136,79,86,290,518,195,84,291,59,277,169,531,100,140,323,197,87,641,177,61,90,281,102,148,522,293,142,322,143,91,546,201,32,103,264,530,106,387,209,297,523,48,154,93,150,268,164,537,769,533,325,114,151,166,107,305,56,578,276,549,270,386,155,80,225,167,109,115,579,170,60,196,329,271,278,157,547,88,117,389,520,337,292,279,198,642,62,178,171,553,199,282,144,63,524,92,173,295,585,202,393,104,645,121,294,283,532,353,179,203,561,324,298,94,581,152,210,181,526,285,108,401,770,95,205,299,527,327,156,534,306,110,326,643,211,116,185,539,168,301,593,226,388,330,548,111,417,771,158,272,213,551,307,118,649,172,227,331,609,338,390,535,309,217,550,159,391,538,119,280,333,580,174,122,777,339,180,554,200,657,354,284,541,313,449,582,229,394,773,123,175,673,555,785,395,562,204,182,341,583,557,64,296,286,233,125,183,212,287,300,355,206,402,186,397,587,345,594,647,528,96,563,241,644,207,705,328,586,403,536,214,357,308,419,565,801,589,187,302,646,112,569,303,595,361,228,650,772,418,540,215,405,310,189,450,332,218,610,597,552,651,160,230,120,334,409,311,421,219,542,369,658,340,774,231,392,556,176,833,314,221,611,653,335,451,778,124,543,342,601,425,315,659,234,775,184,613,356,674,126,396,584,235,558,317,564,661,242,779,559,343,346,617,675,897,404,288,398,188,453,433,208,786,237,588,566,665,781,347,362,127,243,358,706,406,677,590,216,399,567,304,598,359,802,420,625,457,787,349,245,570,190,591,411,648,220,707,312,596,363,803,465,681,407,789,422,599,249,571,370,652,191,835,410,709,232,614,316,573,365,660,423,336,222,689,452,793,371,612,426,602,236,805,413,654,481,834,223,713,344,603,454,544,318,655,427,615,373,776,434,560,238,899,618,348,809,244,721,455,666,319,605,377,662,429,780,239,676,360,837,400,626,246,663,678,435,568,458,817,350,619,459,788,408,782,898,128,667,364,592,466,621,437,737,247,679,572,250,682,351,627,461,708,841,412,783,366,790,441,600,424,669,629,467,901,251,849,804,372,791,192,710,482,574,690,414,683,367,604,794,469,714,374,711,253,575,656,415,807,483,685,905,375,616,224,633,806,428,795,865,430,691,456,715,836,473,810,378,606,620,436,664,722,320,797,431,693,913,240,607,717,485,811,379,839,668,438,818,622,460,838,723,248,697,381,738,680,462,813,628,489,900,842,442,623,929,352,725,784,470,630,819,439,670,463,739,684,252,843,850,497,902,821,729,468,634,903,368,671,792,443,845,631,475,686,851,484,692,712,380,866,796,416,825,961,741,254,906,635,445,694,745,486,808,687,907,471,576,716,376,867,914,474,853,637,798,432,695,812,487,753,724,799,490,857,909,255,718,608,931,477,698,726,915,440,820,840,869,493,719,699,382,814,624,815,499,873,740,930,672,917,822,464,727,962,844,491,632,730,701,383,742,846,921,472,823,852,731,498,881,743,444,904,826,501,688,446,636,933,847,963,733,937,476,854,827,447,746,868,638,800,908,488,696,747,829,858,754,855,916,965,478,910,720,505,749,945,639,870,911,700,859,479,755,918,871,256,728,494,874,932,969,702,816,861,500,757,923,492,732,824,875,977,919,934,744,761,882,495,703,922,828,877,935,848,503,734,883,938,384,964,748,925,993,506,856,735,502,830,966,939,885,946,750,860,756,480,967,831,889,941,751,872,507,971,912,758,947,970,862,978,448,876,920,640,759,949,863,762,509,973,924,878,953,887,936,979,763,504,884,704,994,926,879,981,942,765,496,927,886,995,736,940,985,968,890,948,832,508,943,752,974,997,891,950,760,893,972,1001,954,510,864,982,951,975,764,1009,980,880,955,987,996,892,928,511,766,957,998,983,888,986,944,999,1002,767,989,952,1005,894,1011,958,976,512,1003,895,984,1010,956,988,1013,959,1000,1006,990,1017,991,1012,768,1004,1015,1007,1018,896,1014,992,1019,960,1021,1016,1008,1020,1022,1023,1024]。

n＝1024，排序序列为q序列，记为序列⑨:

[1,2,3,5,9,17,33,4,6,65,7,10,11,129,18,13,19,34,257,21,35,25,66,37,67,513,8,12,130,41,69,14,20,131,49,15,73,258,22,133,36,81,23,259,26,137,38,261,97,145,27,514,39,515,265,68,42,29,161,70,43,517,50,75,273,193,521,529,71,45,132,16,74,289,545,51,24,53,134,82,321,77,135,28,57,83,138,260,40,98,385,30,85,139,262,146,31,99,516,44,89,141,263,147,72,266,46,101,162,518,577,519,52,149,522,47,76,267,163,274,523,641,105,54,153,78,269,194,165,275,55,84,530,525,531,136,58,113,195,290,79,169,277,59,86,61,87,140,100,197,90,291,281,546,769,142,177,91,102,148,143,533,547,537,32,103,264,106,322,293,201,323,578,150,297,48,93,209,549,579,268,386,325,305,164,151,107,56,154,225,387,329,270,114,520,553,642,80,166,109,155,276,60,271,115,167,524,581,561,157,196,170,198,171,278,292,279,117,88,62,178,282,526,643,532,527,63,92,585,770,144,389,294,199,104,337,173,283,202,121,179,295,94,534,645,535,593,548,771,152,324,393,298,203,108,285,181,210,538,95,205,299,353,649,609,326,185,211,401,306,301,110,156,327,116,111,773,550,657,539,551,168,213,307,330,158,226,118,172,331,227,388,309,217,417,338,159,777,272,580,119,541,554,280,333,390,174,122,200,180,229,339,313,391,123,394,284,64,175,341,449,354,395,233,204,673,555,557,562,582,296,286,182,206,125,785,705,183,287,528,583,644,586,563,300,355,212,402,186,397,345,207,302,801,96,187,241,536,587,565,646,594,328,403,214,357,308,303,833,589,647,112,540,569,595,650,772,228,418,332,215,405,310,189,552,610,897,160,218,361,419,409,369,230,450,421,542,597,651,774,658,311,120,334,219,543,611,176,392,231,314,124,340,335,221,451,425,315,556,601,653,234,775,659,613,235,288,453,126,342,396,778,584,558,674,356,343,564,661,559,617,779,675,184,398,433,317,346,242,208,404,358,786,188,588,566,665,625,781,237,127,243,347,399,706,216,677,787,590,567,648,406,359,304,570,596,362,707,190,457,349,465,802,420,407,245,410,591,681,789,571,598,363,573,220,422,599,652,612,709,602,803,370,793,312,191,654,689,603,411,232,249,365,336,423,614,660,655,316,222,371,426,452,413,481,236,223,344,318,544,373,615,427,776,128,454,238,560,834,805,713,835,662,809,605,618,721,780,434,898,319,455,837,817,676,663,348,429,400,377,244,239,458,435,360,568,619,666,737,899,841,782,626,350,246,459,592,678,621,667,788,572,783,627,679,408,364,351,437,192,247,466,412,461,250,366,600,708,574,669,682,790,804,791,710,683,629,441,690,794,604,575,467,424,372,251,414,367,469,482,374,656,901,806,711,428,415,253,616,849,685,795,714,633,691,807,606,483,473,224,664,836,905,810,715,620,797,693,375,485,430,456,722,607,717,811,865,838,697,723,913,818,378,436,813,320,622,240,431,460,379,668,839,489,438,628,623,462,381,439,352,497,680,725,819,842,670,738,630,468,248,442,463,470,443,252,684,843,739,900,821,729,929,850,671,784,631,792,845,902,686,368,634,712,254,692,741,851,825,903,687,471,416,484,445,376,474,906,796,486,635,745,853,961,866,907,716,694,808,475,798,637,695,255,718,576,799,812,867,432,380,698,914,487,724,909,857,719,477,814,608,490,699,753,840,915,726,869,820,815,440,491,624,382,498,493,464,672,930,844,740,917,822,727,632,701,873,931,921,881,730,383,499,444,446,472,962,933,823,742,846,731,447,688,904,636,826,743,852,847,256,476,733,963,937,827,746,501,488,638,505,800,696,854,908,868,855,747,910,829,965,858,754,720,916,870,478,700,749,639,816,945,755,859,911,492,479,728,918,871,384,494,500,874,702,969,495,861,502,932,919,757,922,875,732,977,934,882,761,824,703,923,744,877,848,935,828,734,883,448,938,503,964,925,993,748,735,856,885,939,506,966,507,830,750,946,860,831,967,756,941,912,872,751,889,480,970,947,862,758,971,509,920,640,876,863,759,949,978,924,973,762,878,979,496,936,704,884,953,763,504,926,879,981,994,886,940,927,765,736,887,995,942,968,985,508,890,948,832,752,943,997,972,891,1001,510,950,974,893,951,864,760,1009,511,980,954,764,975,955,880,982,983,928,996,766,957,888,986,998,987,944,892,999,767,512,989,1002,952,1003,894,976,895,1010,956,1005,1011,958,984,959,988,1013,1000,1017,768,990,1004,991,1006,1012,1007,1014,896,960,1015,1018,1019,992,1021,1008,1016,1020,1022,1023,1024]。

n＝1024，排序序列为z序列，记为序列⑩:

[1,2,3,8,4,9,11,27,5,12,13,28,16,32,36,72,6,15,17,33,20,39,43,77,22,45,51,84,58,92,97,165,7,18,21,41,24,47,53,89,30,57,61,100,70,107,116,176,35,63,76,113,78,124,131,188,85,137,144,203,146,220,227,318,10,23,25,56,31,60,69,105,37,73,64,117,82,126,141,198,42,80,86,132,93,145,147,219,101,151,158,228,177,243,260,355,49,90,98,149,108,159,166,235,123,168,187,255,200,272,276,372,138,194,205,275,218,288,301,403,240,308,315,412,335,429,461,535,14,29,34,71,40,79,83,136,46,87,94,148,102,156,161,231,50,96,104,160,114,174,186,250,125,189,201,273,210,286,297,388,59,109,119,185,129,199,206,282,142,212,214,289,237,307,319,408,157,221,241,310,257,333,338,444,267,349,356,454,384,479,506,590,66,128,139,211,150,213,234,309,171,239,254,325,261,334,352,450,178,258,268,347,283,365,381,466,294,389,405,495,415,520,527,640,190,287,291,378,311,394,410,511,324,422,426,526,460,537,562,668,357,449,462,561,486,575,591,691,512,595,616,696,628,715,744,811,19,38,44,88,48,95,103,167,55,106,118,181,127,193,204,299,65,120,130,202,143,215,217,304,153,222,238,317,256,332,339,427,74,140,152,216,170,233,242,331,175,253,262,345,271,353,368,474,184,270,284,367,293,383,402,505,313,411,418,519,447,529,551,666,81,169,172,251,183,266,274,363,192,285,290,380,305,404,414,514,236,296,312,413,320,430,437,528,351,448,463,557,481,574,588,681,263,322,346,436,366,452,473,565,390,477,493,587,513,596,618,712,393,503,521,615,531,621,649,726,560,663,671,751,679,774,792,851,91,182,191,292,232,306,314,409,252,316,323,431,350,445,464,559,269,348,364,451,382,472,485,586,392,487,510,593,524,617,627,723,295,379,391,484,396,496,515,614,417,522,533,626,558,651,669,750,446,549,564,664,589,675,680,771,608,692,695,794,725,795,803,880,321,395,416,523,428,536,552,652,480,563,576,670,594,678,693,777,482,592,613,690,619,694,722,796,639,727,740,812,759,837,847,906,525,620,638,724,650,730,754,819,674,762,772,846,776,852,857,925,682,775,793,853,818,859,882,931,821,891,893,946,912,956,964,985,26,52,54,99,62,110,112,195,67,115,121,207,134,223,226,340,68,133,135,225,162,244,246,358,164,259,280,373,302,397,406,530,75,154,163,248,179,278,281,385,196,303,327,419,328,434,440,538,209,329,344,438,360,456,470,566,374,475,491,582,494,599,612,746,111,173,180,300,208,330,341,433,229,343,359,455,370,469,488,577,247,362,375,476,398,492,497,597,420,501,509,611,545,637,654,761,265,386,407,499,425,516,532,629,441,546,567,646,579,667,677,773,458,573,584,676,607,689,707,785,634,713,731,806,742,820,840,914,122,197,224,342,245,361,371,471,264,376,399,498,421,507,518,622,279,401,424,517,439,543,556,641,457,568,580,672,600,687,705,778,326,435,443,555,467,578,585,683,489,601,606,697,631,711,721,804,508,609,635,716,648,738,743,823,659,752,763,838,786,855,871,927,337,465,478,598,500,605,625,714,541,633,645,737,655,745,758,834,547,653,660,755,684,767,784,848,702,791,802,865,813,878,887,940,569,688,699,781,717,800,808,873,732,817,828,886,839,895,904,950,764,833,843,900,862,910,917,962,869,922,930,967,939,975,984,1004,155,230,249,377,277,400,423,534,298,432,442,548,459,572,583,706,336,453,468,581,490,602,604,708,504,610,632,729,647,741,747,822,354,483,502,603,540,624,636,739,544,644,656,748,665,760,770,841,554,662,685,769,701,783,799,870,719,807,816,877,830,894,898,949,369,539,542,642,553,658,673,765,571,686,698,780,709,801,810,875,630,704,718,809,733,824,827,888,757,832,844,897,858,909,916,961,657,735,749,826,768,836,850,903,787,854,864,915,874,923,933,970,790,868,879,928,889,936,941,977,905,947,954,982,959,990,992,1012,387,550,570,700,623,710,720,805,643,728,736,825,756,829,845,902,661,753,766,835,782,849,861,913,789,863,872,920,884,932,938,973,703,779,788,860,798,867,876,926,815,881,890,937,901,943,951,981,842,896,908,948,918,957,960,988,929,966,969,994,976,997,999,1013,734,797,814,883,831,892,899,944,856,907,911,953,921,958,968,991,866,919,924,965,934,971,972,998,945,978,980,1000,986,1005,1007,1017,885,935,942,974,952,979,983,1002,955,987,989,1006,995,1008,1010,1019,963,993,996,1009,1001,1011,1014,1020,1003,1015,1016,1021,1018,1022,1023,1024]。

n＝1024，排序序列为q序列，记为序列

[1,2,5,9,3,17,33,7,65,513,4,13,6,19,129,10,34,18,11,25,21,35,257,66,37,8,67,130,12,41,20,133,514,14,69,49,73,258,15,22,131,81,27,36,259,137,39,97,23,517,38,26,68,265,42,145,29,70,261,50,43,161,71,521,45,75,82,132,273,193,16,51,74,53,24,515,135,138,289,77,134,83,28,98,260,40,529,57,139,85,30,146,262,44,321,545,99,141,266,31,89,147,263,72,519,162,46,101,385,149,52,47,577,76,267,54,274,165,194,105,516,163,269,78,153,275,55,525,58,113,136,86,79,290,518,195,84,291,59,277,169,531,100,140,323,197,87,641,177,61,90,281,102,148,522,293,142,322,91,143,546,32,201,103,264,530,106,387,48,297,523,209,154,93,150,268,164,537,769,533,325,151,56,166,107,305,114,578,276,549,270,386,155,80,109,225,167,60,579,170,115,196,329,271,278,157,547,88,117,389,520,337,292,279,198,642,62,178,171,553,92,282,199,63,524,144,173,295,585,202,393,104,645,121,294,283,532,353,179,203,561,324,298,94,581,152,210,181,526,285,108,401,770,95,205,299,527,327,156,534,306,110,326,643,211,116,185,539,168,301,593,226,388,330,548,111,417,771,158,272,213,551,307,118,649,172,227,331,609,338,390,535,309,217,550,159,391,538,119,280,333,580,174,122,777,339,200,554,180,657,354,284,541,313,449,582,229,394,773,123,175,673,555,785,204,562,395,182,341,583,557,64,296,286,233,125,183,206,287,300,355,212,402,186,397,587,345,594,647,528,96,563,241,644,207,705,328,586,403,536,214,357,308,419,565,801,589,187,302,646,112,569,303,595,361,228,650,772,418,540,215,405,310,189,450,332,218,610,597,552,651,160,230,120,334,409,311,421,219,542,369,658,231,774,392,176,556,314,833,340,221,611,653,335,451,778,124,543,342,601,425,234,659,235,775,315,613,126,674,356,184,584,288,558,396,564,661,242,779,559,317,346,617,675,897,404,343,398,188,453,208,433,786,237,588,566,665,781,347,362,127,243,358,706,406,677,590,216,399,567,304,598,359,802,420,625,457,787,349,245,570,190,591,220,648,312,707,363,596,465,803,407,681,422,789,249,599,191,571,410,652,370,835,411,709,232,614,316,573,365,660,423,336,222,689,452,793,371,612,426,602,236,805,413,654,481,834,223,713,344,603,318,544,454,655,427,615,373,776,434,560,238,899,618,348,809,244,721,455,666,319,605,377,662,429,780,239,676,360,837,400,626,246,663,678,435,568,458,817,350,619,459,788,466,782,898,364,667,408,592,128,621,437,737,351,679,572,247,682,250,627,461,708,841,412,783,366,790,441,600,424,669,629,467,901,251,849,804,372,791,192,710,482,574,690,414,683,367,604,794,469,714,374,711,253,575,656,483,807,415,685,905,375,616,224,633,806,428,795,865,430,691,473,715,836,456,810,378,606,620,436,664,722,485,797,320,693,913,431,607,717,240,811,379,839,668,438,818,622,460,838,723,248,697,381,738,680,462,813,628,489,900,842,442,623,929,352,725,784,470,630,819,439,670,463,739,684,252,843,850,497,902,821,729,468,634,903,368,671,792,443,845,631,475,686,851,484,692,712,380,866,796,416,825,961,741,254,906,635,445,694,745,486,808,687,907,471,576,716,376,867,914,474,853,637,798,432,695,812,487,753,724,799,490,857,909,255,718,608,931,477,698,726,915,440,820,840,869,493,719,699,382,814,624,815,499,873,740,930,672,917,822,464,727,962,844,491,632,730,701,383,742,846,921,472,823,852,731,498,881,743,444,904,826,501,688,446,933,847,636,963,733,937,476,854,827,447,746,868,638,908,800,488,696,747,829,858,754,855,916,965,478,910,720,505,749,945,639,755,870,256,911,700,859,479,918,871,728,494,874,969,932,702,816,861,500,757,492,875,732,923,824,977,919,934,744,761,882,495,703,922,877,935,828,848,734,503,883,938,384,964,748,506,856,925,993,735,502,830,966,939,885,946,750,860,756,480,967,831,889,941,751,872,507,971,912,758,947,970,862,978,448,876,920,640,759,949,863,762,509,973,924,878,953,887,936,979,763,504,884,704,994,926,879,981,942,765,496,927,886,995,736,940,985,968,890,948,832,508,943,752,974,997,891,950,760,893,972,1001,954,510,864,982,951,975,764,1009,980,880,955,987,996,892,928,511,766,957,998,983,888,986,944,999,1002,767,989,952,1005,894,1011,958,976,512,1003,895,984,1010,956,988,1013,959,1000,1006,990,1017,991,1012,768,1004,1015,1007,1018,896,1014,992,1019,960,1021,1016,1008,1020,1022,1023,1024]。

n＝1024，排序序列为z序列，记为序列

[1,2,5,11,3,13,8,26,4,16,19,29,12,34,39,71,6,18,14,31,21,40,49,75,20,52,43,83,57,91,100,162,7,17,22,44,25,51,47,86,30,55,61,94,65,107,112,169,36,60,72,111,74,116,127,183,88,129,139,198,150,217,224,332,9,24,27,53,35,58,63,104,37,73,66,114,80,124,133,194,42,67,82,137,90,132,147,208,101,151,159,221,174,244,254,351,48,84,97,143,108,153,164,232,120,167,185,251,195,262,276,371,130,187,201,266,209,284,299,394,234,304,320,417,336,428,461,581,15,28,41,68,32,81,77,131,46,78,89,144,98,157,160,226,56,92,102,154,110,175,182,246,125,173,193,259,206,279,296,392,62,106,122,177,118,184,197,269,141,200,219,286,227,303,321,406,149,218,239,309,248,328,337,431,267,344,368,449,384,482,498,611,70,119,136,202,146,215,223,307,163,230,240,325,255,338,355,451,172,247,265,342,281,361,381,468,294,387,399,484,411,514,528,635,196,272,287,376,317,393,403,506,335,422,424,522,454,542,557,662,353,438,462,547,480,563,588,673,496,590,606,698,625,727,757,841,23,38,45,85,59,93,103,165,54,99,115,176,123,191,204,280,69,117,126,189,140,205,214,300,152,222,236,312,250,334,339,433,79,134,138,213,156,235,228,333,170,243,256,340,270,369,373,471,186,261,283,363,293,383,397,486,314,408,426,508,441,532,551,656,95,158,145,242,181,263,258,357,203,274,288,386,301,395,414,513,212,290,306,410,329,419,447,530,347,442,459,545,479,570,585,687,238,311,341,430,362,463,473,559,375,460,488,577,510,597,618,708,401,502,518,609,538,623,633,740,553,648,664,720,675,772,791,880,109,192,168,273,210,291,297,405,231,318,327,435,345,448,469,561,252,343,359,446,382,465,492,579,396,500,504,595,524,616,630,723,277,379,364,475,398,494,512,601,421,520,536,638,555,641,659,747,452,540,566,651,583,667,693,765,599,684,711,802,730,807,817,912,315,385,415,516,450,534,549,646,477,568,572,670,592,678,695,783,490,574,604,705,621,690,737,795,643,743,714,814,761,832,845,897,526,613,628,717,654,733,750,823,681,754,787,858,769,849,869,938,701,799,776,856,805,888,877,929,835,883,904,949,920,961,975,993,10,33,76,121,50,135,105,211,64,155,171,225,128,249,257,350,87,166,142,237,180,260,292,360,178,298,268,380,313,400,418,533,96,161,207,275,190,295,282,390,220,308,323,407,331,434,440,541,241,326,352,436,365,456,470,567,372,481,499,587,509,614,626,738,113,188,199,302,245,316,330,432,229,358,346,455,367,467,483,580,271,348,374,489,389,472,497,600,420,521,531,619,552,649,660,759,289,388,412,519,427,507,537,634,443,544,571,650,582,669,685,774,476,562,591,680,603,691,713,788,636,706,729,810,745,820,838,915,148,216,264,354,233,370,349,485,285,377,391,501,413,525,535,627,310,402,423,511,437,554,564,652,457,550,578,666,602,694,709,780,322,429,444,558,466,565,586,677,493,589,617,697,631,715,735,806,515,615,642,718,657,731,748,824,674,762,771,843,790,853,870,931,356,464,487,593,505,612,624,719,529,622,644,739,661,758,770,834,548,653,672,752,688,763,784,848,704,789,798,860,812,876,887,942,584,676,696,778,726,792,801,866,732,818,825,882,836,894,902,951,751,828,839,896,857,907,916,956,867,919,928,966,937,976,985,1008,179,253,278,378,319,404,425,539,305,416,439,556,458,575,596,689,324,453,478,573,495,598,610,710,517,620,639,722,655,746,753,822,366,474,491,608,523,637,629,734,546,647,663,749,679,773,775,854,569,668,692,766,703,782,796,862,724,804,816,874,826,889,899,948,409,527,503,645,560,671,665,767,594,683,699,786,712,793,809,875,607,700,716,797,744,815,829,884,755,827,844,895,855,910,918,962,640,721,741,819,768,840,847,903,777,850,859,913,872,923,934,969,800,868,878,930,892,940,925,980,900,946,954,973,957,989,995,1013,445,576,543,682,605,702,707,803,632,728,736,821,756,833,842,906,658,742,764,830,781,846,864,914,794,871,861,922,885,933,939,974,686,779,760,852,808,865,873,926,813,879,891,943,901,936,950,982,837,893,908,947,917,955,964,987,924,960,970,998,977,991,1001,1017,725,785,811,881,831,890,898,945,851,909,905,958,921,952,965,992,863,911,927,968,935,963,979,996,944,981,971,999,986,1004,1006,1015,886,932,941,972,953,978,983,1002,959,984,994,1009,988,1003,1011,1020,967,997,990,1007,1000,1014,1010,1019,1005,1012,1016,1021,1018,1022,1023,1024]。

n＝1024，排序序列为q序列，记为序列

[1,2,5,9,3,17,33,7,65,513,4,13,6,19,129,10,34,18,11,25,21,35,257,66,37,8,67,130,12,41,20,133,514,14,69,49,73,258,15,22,131,81,27,36,259,137,39,97,23,517,38,26,68,265,42,145,29,70,261,50,75,161,43,521,71,45,82,132,273,193,16,51,74,24,53,515,135,138,289,77,134,83,28,98,260,40,529,57,139,85,30,146,262,44,321,545,99,141,266,31,89,147,263,72,519,162,46,101,385,149,52,47,577,76,267,54,274,165,194,105,516,163,269,78,153,275,55,525,58,113,136,86,79,290,518,195,84,291,59,277,169,531,100,140,323,197,87,641,177,61,90,281,102,148,522,293,142,322,143,91,546,201,32,103,264,530,106,387,48,297,523,209,154,93,150,268,164,537,769,533,325,151,114,166,107,305,56,578,276,549,270,386,155,80,225,167,109,60,579,170,115,196,329,271,278,157,547,88,117,389,520,337,292,279,198,642,62,178,171,553,199,282,144,63,524,92,173,295,585,202,393,104,645,121,294,283,532,353,179,203,561,324,298,94,581,152,210,181,526,285,108,401,770,95,205,299,527,327,156,534,306,110,326,643,211,116,185,539,168,301,593,226,388,330,548,111,417,771,158,272,213,551,307,118,649,172,227,331,609,338,390,535,309,217,550,159,391,538,119,280,333,580,174,122,777,339,200,554,180,657,354,284,541,313,449,582,229,394,773,123,175,673,555,785,204,562,395,182,341,583,557,64,296,286,233,125,183,206,287,300,355,212,402,186,397,587,345,594,647,528,96,563,241,644,207,705,328,586,403,536,214,357,308,419,565,801,589,187,302,646,112,569,303,595,361,228,650,772,418,540,215,405,310,189,450,332,218,610,597,552,651,160,230,120,334,409,311,421,219,542,369,658,231,774,392,176,556,314,833,340,221,611,653,335,451,778,124,543,342,601,425,315,659,234,775,184,613,126,674,356,235,584,396,558,242,564,661,317,779,559,343,346,617,675,897,404,288,398,188,453,208,433,786,237,588,566,665,781,347,362,127,243,358,706,406,677,590,216,399,567,304,598,359,802,420,625,457,787,349,245,570,190,591,220,648,312,707,363,596,465,803,407,681,422,789,249,599,191,571,410,652,370,835,411,709,232,614,316,573,365,660,423,336,222,689,452,793,371,612,426,602,236,805,413,654,481,834,223,713,344,603,318,544,454,655,427,615,373,776,434,560,238,899,618,348,809,244,721,455,666,319,605,377,662,429,780,239,676,360,837,400,626,246,663,678,435,568,458,817,350,619,459,788,466,782,898,364,667,408,592,128,621,437,737,247,679,572,250,682,351,627,461,708,841,412,783,366,790,441,600,424,669,629,467,901,251,849,804,372,791,192,710,482,574,690,414,683,367,604,794,469,714,374,711,253,575,656,415,807,483,685,905,375,616,224,633,806,428,795,865,430,691,473,715,836,456,810,378,606,620,436,664,722,320,797,431,693,913,240,607,717,485,811,379,839,668,438,818,622,460,838,723,462,697,248,738,680,381,813,628,489,900,842,442,623,929,352,725,784,470,630,819,439,670,463,739,684,252,843,850,497,902,821,729,468,634,903,368,671,792,443,845,631,475,686,851,484,692,712,380,866,796,416,825,961,741,254,906,635,445,694,745,486,808,687,907,471,576,716,376,867,914,474,853,637,798,432,695,812,487,753,724,799,490,857,909,255,718,608,931,477,698,726,915,440,820,840,869,493,719,699,382,814,624,815,499,873,740,930,672,917,822,464,727,962,844,491,632,730,701,383,742,846,921,472,823,852,731,498,881,743,444,904,826,501,688,446,636,933,847,963,733,937,476,854,827,447,746,868,638,908,488,696,747,829,858,754,855,916,965,478,910,720,505,749,945,800,639,870,911,256,755,700,859,479,918,871,728,494,874,932,969,702,816,861,500,757,492,732,923,824,875,977,919,934,744,761,882,495,703,922,877,935,828,848,734,503,883,938,384,964,748,506,856,925,993,735,502,830,966,939,885,946,750,860,756,480,967,831,889,941,751,872,507,971,912,758,947,970,862,978,448,876,920,640,759,949,863,762,509,973,924,878,953,887,936,979,763,504,884,704,994,926,879,981,942,765,496,927,886,995,736,940,985,968,890,948,832,508,943,752,974,997,891,950,760,893,972,1001,954,510,864,982,951,975,764,1009,980,880,955,987,996,892,928,511,766,957,998,983,888,986,944,999,1002,767,989,952,1005,894,1011,958,976,512,1003,895,984,1010,956,988,1013,959,1000,1006,990,1017,991,1012,768,1004,1015,1007,1018,896,1014,992,1019,960,1021,1016,1008,1020,1022,1023,1024]。

n＝1024，排序序列为z序列，记为序列

[1,2,5,11,3,13,8,26,4,16,19,29,12,34,39,71,6,18,14,31,21,40,49,74,20,52,43,83,57,91,100,163,7,17,22,44,25,51,47,86,30,55,63,94,66,107,112,169,36,60,72,111,75,116,127,187,88,129,139,198,150,217,224,332,9,24,27,53,35,58,65,104,37,73,61,114,80,124,133,194,42,67,82,137,90,132,147,208,101,151,160,226,174,244,254,351,48,84,97,143,108,153,164,232,120,167,185,251,197,262,276,371,130,183,201,266,209,284,299,394,234,304,320,417,336,428,461,581,15,28,41,68,32,81,77,131,46,78,89,144,98,157,159,223,56,92,102,154,110,175,182,246,125,173,193,259,206,279,296,392,62,106,122,177,118,184,196,269,141,200,219,286,227,303,321,406,149,218,239,309,248,328,337,426,267,344,368,449,384,482,498,611,70,119,136,202,146,215,221,307,162,230,240,325,255,338,355,451,172,247,265,342,281,361,381,468,294,387,399,484,411,514,528,635,195,272,287,376,317,393,403,506,335,424,431,522,454,542,557,659,353,435,462,547,480,563,585,675,496,588,606,698,625,727,757,841,23,38,45,85,59,93,103,165,54,99,115,176,123,191,204,280,69,117,126,189,140,205,214,300,152,222,236,312,250,334,339,447,79,134,138,213,156,235,228,333,170,243,256,340,270,369,373,471,186,261,283,363,293,383,397,486,314,408,422,508,438,532,551,654,95,158,145,242,181,263,258,357,203,274,288,386,301,395,414,513,212,290,306,410,329,419,441,530,347,442,459,545,479,570,590,687,238,311,341,430,362,463,473,559,375,460,488,577,510,597,618,708,401,502,518,609,538,623,633,740,553,648,664,720,678,772,791,880,109,192,168,273,210,291,297,405,231,318,327,433,345,448,469,561,252,343,359,446,382,465,492,579,396,500,504,595,524,616,628,723,277,379,364,475,398,494,512,601,421,520,536,638,555,641,656,747,452,540,566,651,583,667,693,765,599,684,711,802,730,807,817,912,315,385,415,516,450,534,549,646,477,568,572,670,592,673,695,783,490,574,604,705,621,690,737,795,643,743,714,814,761,831,845,897,526,613,630,717,662,733,750,822,681,754,787,858,769,849,869,938,701,799,776,856,805,888,877,929,834,883,904,949,920,961,975,993,10,33,76,121,50,135,105,211,64,155,171,225,128,249,257,350,87,166,142,237,180,260,292,360,178,298,268,380,313,400,418,533,96,161,207,275,190,295,282,390,220,308,323,407,331,434,440,541,241,326,352,436,365,456,470,567,372,481,499,587,509,614,626,738,113,188,199,302,245,316,330,432,229,358,346,455,367,467,483,580,271,348,374,489,389,472,497,600,420,521,531,619,552,649,660,759,289,388,412,519,427,507,537,634,443,544,571,650,582,669,685,774,476,562,591,680,603,691,713,788,636,706,729,808,745,820,838,915,148,216,264,354,233,370,349,485,285,377,391,501,413,525,535,627,310,402,423,511,437,554,564,652,457,550,578,666,602,694,709,780,322,429,444,558,466,565,586,677,493,589,617,697,631,715,735,806,515,615,642,718,657,731,748,823,674,762,771,843,790,853,870,931,356,464,487,593,505,612,624,719,529,622,644,739,661,758,770,833,548,653,672,752,688,763,784,848,704,789,798,859,812,876,887,942,584,676,696,778,726,792,801,866,732,818,824,882,835,894,902,951,751,827,842,896,857,907,916,956,867,919,928,966,937,976,985,1008,179,253,278,378,319,404,425,539,305,416,439,556,458,575,596,689,324,453,478,573,495,598,610,710,517,620,639,722,655,746,753,837,366,474,491,608,523,637,629,734,546,647,663,749,679,773,775,854,569,668,692,766,703,782,796,861,724,804,816,874,825,889,899,948,409,527,503,645,560,671,665,767,594,683,699,786,712,793,810,875,607,700,716,797,744,815,828,884,755,826,844,895,855,910,918,962,640,721,741,819,768,839,847,903,777,850,862,913,872,923,934,969,800,868,878,930,892,940,925,980,900,946,954,973,957,989,995,1013,445,576,543,682,605,702,707,803,632,728,736,821,756,832,840,906,658,742,764,829,781,846,864,914,794,871,860,922,885,933,939,974,686,779,760,851,809,865,873,926,813,879,891,943,901,936,950,982,836,893,908,947,917,955,964,987,924,960,970,998,977,991,1001,1017,725,785,811,881,830,890,898,945,852,909,905,958,921,952,965,992,863,911,927,968,935,963,979,996,944,981,971,999,986,1004,1006,1015,886,932,941,972,953,978,983,1002,959,984,994,1009,988,1003,1011,1020,967,997,990,1007,1000,1014,1010,1019,1005,1012,1016,1021,1018,1022,1023,1024]。

n＝1024，排序序列为q序列，记为序列

[1,2,5,9,3,17,33,7,65,513,4,13,6,19,129,10,34,18,11,25,21,35,257,66,37,8,67,130,12,41,20,133,514,14,69,49,73,258,15,22,131,81,27,36,259,137,39,97,23,517,38,26,68,265,42,145,29,70,261,50,43,161,71,521,45,75,82,132,273,193,16,51,74,53,24,515,135,138,289,77,134,83,28,98,260,40,529,57,139,85,30,146,262,44,321,545,99,141,266,31,89,147,263,72,519,162,46,101,385,149,52,47,577,76,267,54,274,165,194,105,516,163,269,78,153,275,55,525,58,113,136,86,79,290,518,195,84,291,59,277,169,531,100,140,323,197,87,641,177,61,90,281,102,148,522,293,142,322,91,143,546,32,201,103,264,530,106,387,48,297,523,209,154,93,150,268,164,537,769,533,325,151,56,166,107,305,114,578,276,549,270,386,155,80,109,225,167,60,579,170,115,196,329,271,278,157,547,88,117,389,520,337,292,279,198,642,62,178,171,553,92,282,199,63,524,144,173,295,585,202,393,104,645,121,294,283,532,353,179,203,561,324,298,94,581,152,210,181,526,285,108,401,770,95,205,299,527,327,156,534,306,110,326,643,211,116,185,539,168,301,593,226,388,330,548,111,417,771,158,272,213,551,307,118,649,172,227,331,609,338,390,535,309,217,550,159,391,538,119,280,333,580,174,122,777,339,200,554,180,657,354,284,541,313,449,582,229,394,773,123,175,673,555,785,204,562,395,182,341,583,557,64,296,286,233,125,183,206,287,300,355,212,402,186,397,587,345,594,647,528,96,563,241,644,207,705,328,586,403,536,214,357,308,419,565,801,589,187,302,646,112,569,303,595,361,228,650,772,418,540,215,405,310,189,450,332,218,610,597,552,651,160,230,120,334,409,311,421,219,542,369,658,231,774,392,176,556,314,833,340,221,611,653,335,451,778,124,543,342,601,425,234,659,235,775,315,613,126,674,356,184,584,288,558,396,564,661,242,779,559,317,346,617,675,897,404,343,398,188,453,208,433,786,237,588,566,665,781,347,362,127,243,358,706,406,677,590,216,399,567,304,598,359,802,420,625,457,787,349,245,570,190,591,220,648,312,707,363,596,465,803,407,681,422,789,249,599,191,571,410,652,370,835,411,709,232,614,316,573,365,660,423,336,222,689,452,793,371,612,426,602,236,805,413,654,481,834,223,713,344,603,318,544,454,655,427,615,373,776,434,560,238,899,618,348,809,244,721,455,666,319,605,377,662,429,780,239,676,360,837,400,626,246,663,678,435,568,458,817,350,619,459,788,466,782,898,364,667,408,592,128,621,437,737,351,679,572,247,682,250,627,461,708,841,412,783,366,790,441,600,424,669,629,467,901,251,849,804,372,791,192,710,482,574,690,414,683,367,604,794,469,714,374,711,253,575,656,483,807,415,685,905,375,616,224,633,806,428,795,865,430,691,473,715,836,456,810,378,606,620,436,664,722,485,797,320,693,913,431,607,717,240,811,379,839,668,438,818,622,460,838,723,248,697,381,738,680,462,813,628,489,900,842,442,623,929,352,725,784,470,630,819,439,670,463,739,684,252,843,850,497,902,821,729,468,634,903,368,671,792,443,845,631,475,686,851,484,692,712,380,866,796,416,825,961,741,254,906,635,445,694,745,486,808,687,907,471,576,716,376,867,914,474,853,637,798,432,695,812,487,753,724,799,490,857,909,255,718,608,931,477,698,726,915,440,820,840,869,493,719,699,382,814,624,815,499,873,740,930,672,917,822,464,727,962,844,491,632,730,701,383,742,846,921,472,823,852,731,498,881,743,444,904,826,501,688,446,933,847,636,963,733,937,476,854,827,447,746,868,638,908,800,488,696,747,829,858,754,855,916,965,478,910,720,505,749,945,639,755,870,256,911,700,859,479,918,871,728,494,874,969,932,702,816,861,500,757,492,875,732,923,824,977,919,934,744,761,882,495,703,922,877,935,828,848,734,503,883,938,384,964,748,506,856,925,993,735,502,830,966,939,885,946,750,860,756,480,967,831,889,941,751,872,507,971,912,758,947,970,862,978,448,876,920,640,759,949,863,762,509,973,924,878,953,887,936,979,763,504,884,704,994,926,879,981,942,765,496,927,886,995,736,940,985,968,890,948,832,508,943,752,974,997,891,950,760,893,972,1001,954,510,864,982,951,975,764,1009,980,880,955,987,996,892,928,511,766,957,998,983,888,986,944,999,1002,767,989,952,1005,894,1011,958,976,512,1003,895,984,1010,956,988,1013,959,1000,1006,990,1017,991,1012,768,1004,1015,1007,1018,896,1014,992,1019,960,1021,1016,1008,1020,1022,1023,1024]。

n＝1024，排序序列为z序列，记为序列

[1,2,5,11,3,13,8,26,4,16,19,29,12,34,39,71,6,18,14,31,21,40,49,75,20,52,43,83,57,91,100,162,7,17,22,44,25,51,47,86,30,55,61,94,65,107,112,169,36,60,72,111,74,116,127,183,88,129,139,198,150,217,224,332,9,24,27,53,35,58,63,104,37,73,66,114,80,124,133,194,42,67,82,137,90,132,147,208,101,151,159,221,174,244,254,351,48,84,97,143,108,153,164,232,120,167,185,251,195,262,276,371,130,187,201,266,209,284,299,394,234,304,320,417,336,428,461,581,15,28,41,68,32,81,77,131,46,78,89,144,98,157,160,226,56,92,102,154,110,175,182,246,125,173,193,259,206,279,296,392,62,106,122,177,118,184,197,269,141,200,219,286,227,303,321,406,149,218,239,309,248,328,337,431,267,344,368,449,384,482,498,611,70,119,136,202,146,215,223,307,163,230,240,325,255,338,355,451,172,247,265,342,281,361,381,468,294,387,399,484,411,514,528,635,196,272,287,376,317,393,403,506,335,422,424,522,454,542,557,662,353,438,462,547,480,563,588,673,496,590,606,698,625,727,757,841,23,38,45,85,59,93,103,165,54,99,115,176,123,191,204,280,69,117,126,189,140,205,214,300,152,222,236,312,250,334,339,433,79,134,138,213,156,235,228,333,170,243,256,340,270,369,373,471,186,261,283,363,293,383,397,486,314,408,426,508,441,532,551,656,95,158,145,242,181,263,258,357,203,274,288,386,301,395,414,513,212,290,306,410,329,419,447,530,347,442,459,545,479,570,585,687,238,311,341,430,362,463,473,559,375,460,488,577,510,597,618,708,401,502,518,609,538,623,633,740,553,648,664,720,675,772,791,880,109,192,168,273,210,291,297,405,231,318,327,435,345,448,469,561,252,343,359,446,382,465,492,579,396,500,504,595,524,616,630,723,277,379,364,475,398,494,512,601,421,520,536,638,555,641,659,747,452,540,566,651,583,667,693,765,599,684,711,802,730,807,817,912,315,385,415,516,450,534,549,646,477,568,572,670,592,678,695,783,490,574,604,705,621,690,737,795,643,743,714,814,761,832,845,897,526,613,628,717,654,733,750,823,681,754,787,858,769,849,869,938,701,799,776,856,805,888,877,929,835,883,904,949,920,961,975,993,10,33,76,121,50,135,105,211,64,155,171,225,128,249,257,350,87,166,142,237,180,260,292,360,178,298,268,380,313,400,418,533,96,161,207,275,190,295,282,390,220,308,323,407,331,434,440,541,241,326,352,436,365,456,470,567,372,481,499,587,509,614,626,738,113,188,199,302,245,316,330,432,229,358,346,455,367,467,483,580,271,348,374,489,389,472,497,600,420,521,531,619,552,649,660,759,289,388,412,519,427,507,537,634,443,544,571,650,582,669,685,774,476,562,591,680,603,691,713,788,636,706,729,810,745,820,838,915,148,216,264,354,233,370,349,485,285,377,391,501,413,525,535,627,310,402,423,511,437,554,564,652,457,550,578,666,602,694,709,780,322,429,444,558,466,565,586,677,493,589,617,697,631,715,735,806,515,615,642,718,657,731,748,824,674,762,771,843,790,853,870,931,356,464,487,593,505,612,624,719,529,622,644,739,661,758,770,834,548,653,672,752,688,763,784,848,704,789,798,860,812,876,887,942,584,676,696,778,726,792,801,866,732,818,825,882,836,894,902,951,751,828,839,896,857,907,916,956,867,919,928,966,937,976,985,1008,179,253,278,378,319,404,425,539,305,416,439,556,458,575,596,689,324,453,478,573,495,598,610,710,517,620,639,722,655,746,753,822,366,474,491,608,523,637,629,734,546,647,663,749,679,773,775,854,569,668,692,766,703,782,796,862,724,804,816,874,826,889,899,948,409,527,503,645,560,671,665,767,594,683,699,786,712,793,809,875,607,700,716,797,744,815,829,884,755,827,844,895,855,910,918,962,640,721,741,819,768,840,847,903,777,850,859,913,872,923,934,969,800,868,878,930,892,940,925,980,900,946,954,973,957,989,995,1013,445,576,543,682,605,702,707,803,632,728,736,821,756,833,842,906,658,742,764,830,781,846,864,914,794,871,861,922,885,933,939,974,686,779,760,852,808,865,873,926,813,879,891,943,901,936,950,982,837,893,908,947,917,955,964,987,924,960,970,998,977,991,1001,1017,725,785,811,881,831,890,898,945,851,909,905,958,921,952,965,992,863,911,927,968,935,963,979,996,944,981,971,999,986,1004,1006,1015,886,932,941,972,953,978,983,1002,959,984,994,1009,988,1003,1011,1020,967,997,990,1007,1000,1014,1010,1019,1005,1012,1016,1021,1018,1022,1023,1024]。

需要说明的是，上述一些排序序列只是一些举例，其应用到polar编码过程中会有助于提高poalr码编译码性能。任一种举例的排序序列中，在不影响其整体效果的前提下，可以做包括但不限于如下几方面的调整或者等同替换：

1、排序序列中少数元素之间的位置互换。例如，序号位置可以在设定幅度内调整，例如，设定幅度为5，将序号为10的元素位置在左右5个位置内调整均可；

2、排序序列中的部分元素进行调整，但是根据该排序序列选择出用于传输t比特信息的信道集合是一致或相似的。

3、排序序列中包含从1开始到n结束的n个元素，从1开始到n结束的n个元素代表n个极化信道的序号。实际上，n个极化信道的序号也可以从0开始到n-1结束，将上述排序序列中各序号均减1即可，这也是上述各计算方式中的序号形式。当然，也可以采取其他方式表示上述极化信道的序号或者标识，该具体表达方式不影响排序序列中所表示的极化信道的具体位置。

4、上述排序序列中的n个极化信道的序号是按照n个极化信道可靠度从高到低排列的，实际上，n个极化信道的序号也可以按照n个极化信道可靠度从小到大排列，将上述排序序列中的元素逆序排列或者反序排列即可。

5、上述排序序列还可以利用各个信道的归一化可靠度或等效可靠度排序序列进行表征。例如：信道x在上述序列的排序位置为n(排序序列最小序号记为1)，则该信道的可靠度可以表示为n或者归一化的n/n，n为排序序列的长度。

基于图2所示的polar码编码方法，如图3所示，本申请实施例还提供了一种polar码编码装置300，polar码编码装置300用于执行图2所示的polar码编码方法，polar码编码装置300包括：

确定单元301，用于确定用于对待编码比特进行编码的排序序列，排序序列用于表征n个极化信道的可靠度排序，n为polar码的母码长度，n为2的正整数次幂；

编码单元302，用于采用排序序列对待编码比特进行polar码编码，得到编码后的比特。

可选的，待编码比特长度为k，k个待编码比特被映射到n个极化信道中的k个极化信道上，k个极化信道的可靠度高于剩余的n-k个极化信道的可靠度。

可选的，确定单元301还用于：

获取第一序列，第二序列，第三序列，第四序列和第五序列；

若排序序列按照极化信道的可靠度由低到高的顺序排列，则确定单元301具体用于：

获取第二序列中序号小于等于64的第一子序列，按照第一子序列的前(64-t1+1)个序号在第一序列中的排序，更新第一子序列的前(64-t1+1)个序号在第一子序列中的排序，基于更新后的第一子序列获得待对比第二序列；基于第一目标长度k1，分别在第二序列和待对比第二序列中选择k1个最可靠位置对应的序号，分别基于选择的序号和已生成的第二中间结果序列获得编码码字性能结果，其中，第二中间结果序列中包含已选择的(k1-1)个序号，目标长度k1依次取值从t1到kmax^(m)，kmax^(m)为母码长度为m时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新第二中间结果序列；将k1取值为kmax^(m)时获得的更新后的第二中间结果序列作为更新后第一子序列，将更新后第一子序列置于第二序列中，获得更新后第二序列；

获取第三序列中序号小于等于128的第二子序列，按照第二子序列的前(128-t2+1)个序号在更新后第二序列中的排序，更新第二子序列的前(128-t2+1)个序号在第二子序列中的排序，基于更新后的第二子序列获得待对比第三序列；基于第二目标长度k2，分别在第三序列和待对比第三序列中选择k2个最可靠位置对应的序号，分别基于选择的序号和已生成的第三中间结果序列获得编码码字性能结果，其中，第三中间结果序列中包含已选择的(k2-1)个序号，第二目标长度k2依次取值从t2到kmax^(m)，kmax^(m)为母码长度为m＝128时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新第三中间结果序列；将k2取值为kmax^(m)时获得的更新后的第三中间结果序列作为更新后第二子序列，将更新后第二子序列置于第三序列中，获得更新后第三序列；

获取第四序列中序号小于等于256的第三子序列，按照第三子序列的前(256-t3+1)个序号在更新后第三序列中的排序，更新第三子序列的前(256-t3+1)个序号在第三子序列中的排序，基于更新后的第三子序列获得待对比第四序列；基于第三目标长度k3，分别在第四序列和待对比第四序列中选择k3个最可靠位置对应的序号，分别基于选择的序号和已生成的第四中间结果序列获得编码码字性能结果，其中，第四中间结果序列中包含已选择的(k3-1)个序号，第三目标长度k3依次取值从t3到kmax^(m)，kmax^(m)为母码长度为m＝256时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新第四中间结果序列；将k3取值为kmax^(m)时获得的更新后的第四中间结果序列作为更新后第三子序列，将更新后第三子序列置于第四序列中，获得更新后第四序列；

获取第五序列中序号小于等于512的第四子序列，按照第四子序列的前(512-t4+1)个序号在更新后第四序列中的排序，更新第四子序列的前(512-t4+1)个序号在第四子序列中的排序，基于更新后的第四子序列获得待对比第五序列；基于第四目标长度k4，分别在第五序列和待对比第五序列中选择k4个最可靠位置对应的序号，分别基于选择的序号和已生成的第五中间结果序列获得编码码字性能结果，其中，第五中间结果序列中包含已选择的(k4-1)个序号，第四目标长度k4依次取值从t4到kmax^(m)，kmax^(m)为母码长度为m＝512时的最大信息比特长度；按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新第五中间结果序列；将k4取值为kmax^(m)时获得的更新后的第五中间结果序列作为更新后第四子序列，将更新后第四子序列置于第五序列中，获得更新后第五序列；

其中，t1、t2、t3和t4由性能优劣判断规则确定。

可选的，确定单元301在获取第一序列，第二序列，第三序列，第四序列和第五序列时，具体用于：

根据第五目标长度k5，在(m-k5+1)个序号中依次选择一个序号，每选择一个序号，基于所选择的一个序号与已生成的中间结果序列获得编码码字性能结果，其中，中间结果序列包含已选择的(k5-1)个序号，(m-k5+1)个序号与中间结果序列不相交，目标长度k5依次取值从1到kmax^(m)，kmax^(m)为母码长度为m时的最大信息比特长度；

按照性能优劣判断规则，选择获得最优性能的序号，按照所选择的获得最优性能的序号更新中间结果序列；

其中，m为2的整数次幂，在m＝64时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为第一序列；在m＝128时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为第二序列；在m＝256时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为第三序列；在m＝512时，将k5取值为kmax时获得的更新后的中间结果序列作为第四序列；在m＝1024时，将k5取值为kmax^(m)时获得的更新后的中间结果序列作为第五序列。

基于图2所示的polar码编码方法的同一发明构思，如图4所示，本申请实施例中还提供一种polar码编码装置400，该polar码编码装置400用于执行图2所示的polar码编码方法。图2所示的polar码编码方法中的部分或全部可以通过硬件来实现也可以通过软件来实现，当通过硬件实现时，polar码编码装置400包括：输入接口电路401，用于获取待编码比特；逻辑电路402，用于执行上述图2所示的polar码编码方法，具体请见前面方法实施例中的描述，此处不再赘述；输出接口电路403，用于输出编码后的比特序列。

可选的，polar码编码装置400在具体实现时可以是芯片或者集成电路。

可选的，当上述实施例的polar码编码方法中的部分或全部通过软件来实现时，如图5所示，polar码编码装置400包括：存储器501，用于存储程序；处理器502，用于执行存储器501存储的程序，当程序被执行时，使得polar码编码装置400可以实现上述图2实施例提供的polar码编码方法。

可选的，上述存储器501可以是物理上独立的单元，也可以如图6所示，存储器501与处理器502集成在一起。

可选的，当上述图2实施例的编码方法中的部分或全部通过软件实现时，polar码传输装置400也可以只包括处理器502。用于存储程序的存储器501位于polar码传输装置400之外，处理器502通过电路/电线与存储器501连接，用于读取并执行存储器501中存储的程序。

处理器502可以是中央处理器(centralprocessingunit，cpu)，网络处理器(networkprocessor，np)或者cpu和np的组合。

处理器502还可以进一步包括硬件芯片。上述硬件芯片可以是专用集成电路(application-specificintegratedcircuit，asic)，可编程逻辑器件(programmablelogicdevice，pld)或其组合。上述pld可以是复杂可编程逻辑器件(complexprogrammablelogicdevice，cpld)，现场可编程逻辑门阵列(field-programmablegatearray，fpga)，通用阵列逻辑(genericarraylogic,gal)或其任意组合。

存储器501可以包括易失性存储器(volatilememory)，例如随机存取存储器(random-accessmemory，ram)；存储器501也可以包括非易失性存储器(non-volatilememory)，例如快闪存储器(flashmemory)，硬盘(harddiskdrive，hdd)或固态硬盘(solid-statedrive，ssd)；存储器501还可以包括上述种类的存储器的组合。

本申请实施例还提供了一种计算机存储介质，存储有计算机程序，该计算机程序包括用于执行图2所示的polar码编码方法。

本申请实施例还提供了一种包含指令的计算机程序产品，当其在计算机上运行时，使得计算机执行图2所示的polar码编码方法。

本领域内的技术人员应明白，本申请的实施例可提供为方法、系统、或计算机程序产品。因此，本申请可采用完全硬件实施例、完全软件实施例、或结合软件和硬件方面的实施例的形式。而且，本申请可采用在一个或多个其中包含有计算机可用程序代码的计算机可用存储介质(包括但不限于磁盘存储器、cd-rom、光学存储器等)上实施的计算机程序产品的形式。

本申请是参照根据本申请实施例的方法、设备(系统)、和计算机程序产品的流程图和/或方框图来描述的。应理解可由计算机程序指令实现流程图和/或方框图中的每一流程和/或方框、以及流程图和/或方框图中的流程和/或方框的结合。可提供这些计算机程序指令到通用计算机、专用计算机、嵌入式处理机或其他可编程数据处理设备的处理器以产生一个机器，使得通过计算机或其他可编程数据处理设备的处理器执行的指令产生用于实现在流程图一个流程或多个流程和/或方框图一个方框或多个方框中指定的功能的装置。

这些计算机程序指令也可存储在能引导计算机或其他可编程数据处理设备以特定方式工作的计算机可读存储器中，使得存储在该计算机可读存储器中的指令产生包括指令装置的制造品，该指令装置实现在流程图一个流程或多个流程和/或方框图一个方框或多个方框中指定的功能。

这些计算机程序指令也可装载到计算机或其他可编程数据处理设备上，使得在计算机或其他可编程设备上执行一系列操作步骤以产生计算机实现的处理，从而在计算机或其他可编程设备上执行的指令提供用于实现在流程图一个流程或多个流程和/或方框图一个方框或多个方框中指定的功能的步骤。

尽管已描述了本申请的优选实施例，但本领域内的技术人员一旦得知了基本创造性概念，则可对这些实施例作出另外的变更和修改。所以，所附权利要求意欲解释为包括优选实施例以及落入本申请范围的所有变更和修改。

显然，本领域的技术人员可以对本申请实施例进行各种改动和变型而不脱离本申请实施例的精神和范围。这样，倘若本申请实施例的这些修改和变型属于本申请权利要求及其等同技术的范围之内，则本申请也意图包含这些改动和变型在内。