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。

本申请实施例提供的具体技术方案如下：

第一方面，提供一种polar码编码方法，获取待编码的比特，所述待编码的比特长度为k，k为正整数，获取用于对k个待编码比特进行编码的序列，记为第一序列，所述第一序列用于表征n个极化信道的可靠度的排序，所述第一序列中包含n个极化信道的序号，所述n个极化信道的序号在所述第一序列中是按照所述n个极化信道的可靠度进行排列的，n为polar码的母码长度，n为2的正整数次幂，k≤n，按照可靠度从高到低的顺序，从所述第一序列中选出可靠度排序较高的前k个序号，将待编码的信息比特映射到所述前k个序号的极化信道上，并对所述待编码比特进行polar码编码。从而实现了通过polar码极化信道的可靠度的计算来确定信息比特与固定比特的位置，与信道参数及码率没有关系，可以降低polar码编码的计算复杂度。

在一个可能的设计中，获取n个极化信道中每一个极化信道的可靠度，其中，n个极化信道中第i个极化信道的可靠度通过以下任一种可能的设计中的可靠度计算公式确定。可选的，极化信道的可靠度可以在线计算，也可以离线计算并存储。由于这类构造序列的方式可以做到与物理信道参数无关，因此便于离线存储。

在一个可能的设计中，所述n个极化信道中第i个极化信道的可靠度通过第一可靠度计算公式确定；所述第一可靠度计算公式为：其中，1≤i≤n，wi为第i个极化信道的可靠度，n＝log2n，i满足i-1＝bn-1bn-2...b0，其中bn-1bn-2...b1b0为i-1的二进制表示，bj为信道序号i的二进制表示中第j位的取值,bj∈{0,1}，ξ为所述第一可靠度计算公式包含的ξ阶数的集合，c(ξ,j)为ξ和j的函数、表征各阶项的权重值，ψ(ξ，j)为ξ和j的函数，e(ξ)为ξ的函数、表征ξ阶核函数ψ(ξ，j)随j变化的频率。

在一个可能的设计中，所述n个极化信道中第i个极化信道的可靠度通过第二可靠度计算公式确定；所述第二可靠度计算公式为：其中，1≤i≤n，wi为第i个极化信道的可靠度，n＝log2n，i满足i-1＝bn-1bn-2…b0，其中bn-1bn-2...b1b0为i-1的二进制表示，bj为信道序号i的二进制表示中第j位的取值，bj∈{0,1}，ξ为所述第二可靠度计算公式包含的ξ阶数的集合，c(ξ)为ξ的函数、表征各阶项的权重值，β为大于1的常量，e(ξ)为ξ的函数、表征ξ阶β核随j变化的频率。

在一个可能的设计中，所述n个极化信道中第i个极化信道的可靠度通过第三可靠度计算公式确定；所述第三可靠度计算公式为：其中，其中，1≤i≤n，wi为第i个极化信道的可靠度，n＝log2n，i满足i-1＝bn-1bn-2...b0，其中bn-1bn-2...b1b0为i-1的二进制表示，bj为信道序号i的二进制表示中第j位的取值，bj∈{0,1}，ξ为所述第三可靠度计算公式包含的ξ阶数的集合，c(ξ)为ξ的函数、表征各阶项的权重值，β为大于1的常量。

在一个可能的设计中，所述n个极化信道中第i个极化信道的可靠度通过第四可靠度计算公式确定；所述第四可靠度计算公式为：其中，其中，1≤i≤n，wi为第i个极化信道的可靠度，n＝log2n，i满足i-1＝bn-1bn-2...b0，其中bn-1bn-2...b1b0为i-1的二进制表示，bj为信道序号i的二进制表示中第j位的取值,bj∈{0,1}，ξ1和ξ2为所述第四可靠度计算公式的ξ阶数的集合，c(ξ)为ξ的函数、表征各阶项的权重值，ψ(ξ，j)为ξ和j的函数，β为大于1的常量，e(ξ)为ξ的函数、表征ξ阶β核随j变化的频率或ξ阶核函数ψ(ξ，j)随j变化的频率。

在一个可能的设计中，所述n个极化信道中第i个极化信道的可靠度通过第五可靠度计算公式确定；所述第五可靠度计算公式为：其中，1≤i≤n，wi为第i个极化信道的可靠度，n＝log2n，i满足i-1＝bn-1bn-2...b0，其中bn-1bn-2...b1b0为i-1的二进制表示，bj为信道序号i的二进制表示中第j位的取值,bj∈{0,1}，c(j)为常数是各个二进制位核函数前的系数，e为常数描述了核函数ψ(j)随j变化的频率，核函数ψ(j)是信道序号i的二级制表征位j的函数、即各个二进制位的核函数可以不同。

在一个可能的设计中，所述n个极化信道中的第i个极化信道的可靠度通过第六可靠度计算公式确定；所述第六可靠度计算公式为：

其中，1≤i≤n，wi为第i个极化信道的可靠度，n＝log2n，i满足i-1＝bn-1bn-2...b0，其中bn-1bn-2...b1b0为i-1的二进制表示，bj为信道序号i的二进制表示中第j位的取值,bj∈{0,1}，β为大于1的常量，k为一个包含信道序号二进制表达位的集合，即元素包含0到n-1，k为集合{0,1，…，n-1}的子集，其集合大小0≤＝size(k)≤n-1，如果m出现在k集合中，那么信道序号0到2m-1与信道序号2^m到2^m+1-1在序列中的排序就会不一致，k集合中的元素会改变序列的对称特性，其中a、c分别为核函数b、d的权重。

在一个可能的设计中，

在一个可能的设计中，极化信道序号i的取值范围为0≤i≤n-1，此时与上述各种可能的设计相比，上述各种设计中的bn-1bn-2...b1b0为i的二进制表示。

通过以上任一种形式的公式计算n个极化信道中第i个极化信道的可靠度，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。

在一个可能的设计中，所述第一序列为第二序列的全部或者子集，所述第二序列中包含nmax个极化信道的序号，所述nmax个极化信道的序号在所述第二序列中是按照所述nmax个极化信道的可靠度进行排列的，所述nmax为2的正整数次幂，所述nmax≥n，所述nmax为polar码的最大母码长度，例如，所述nmax＝1024，或者，所述nmax＝512。由于序列要符合嵌套特性，因此当最大母码长度的第二序列确定，可以根据所述第二序列获得其它较短母码长度的序列。

在一个可能的设计中，所述第二序列可以通过上述任一种可能的设计中的公式确定，其中，将公式中的n替换为nmax即可。

在一个可能的设计中，当a＝-0.22，b＝1.2，c＝0.26，d＝1，f＝0，g＝1且ξ＝{0，1}时，第一可靠度计算公式可以具体化为利用具体化后的该公式计算极化信道的可靠度，当n＝1024，所述第一序列为以下序列，或者，当nmax＝1024，所述第二序列也可以为以下序列，极化信道序号从1开始：

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在一个可能的设计中，ξ＝{0，1}包含0阶项及1阶项、系数定义为时，第三可靠度计算公式可以具体化为利用具体化后的该公式计算极化信道的可靠度，当n＝1024，所述第一序列为以下序列，或者，当nmax＝1024，所述第二序列也可以为以下序列，极化信道序号从1开始：

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

在一个可能的设计中，ξ＝{0，1，2}包含0阶项、1阶项及2阶项，系数定义为时，第三可靠度计算公式可以具体化为利用具体化后的该公式计算极化信道的可靠度，当n＝1024，所述第一序列为以下序列，或者，当nmax＝1024，所述第二序列也可以为以下序列，极化信道序号从1开始：

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

在一个可能的设计中，对于ξ仅属于ξ1的集合，其核函数为β，对于例如：当ξ＝1时，c(1)＝1，ψ(1)＝β＝2^1/4＝1.1892，e(1)＝1；当ξ＝2时，c(2)＝0.17，ψ(2)＝0.63，e(1)＝1/4，或者c(2)＝0.17，ψ(2)＝0.8909，e(1)＝1；第四可靠度计算公式可以具体化为或者，利用具体化后的该公式计算极化信道的可靠度，当n＝1024，所述第一序列为以下序列，或者，当nmax＝1024，所述第二序列也可以为以下序列，极化信道序号从1开始：

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在一个可能的设计中，所述第一序列或者所述第二序列可以为说明书中序列①至序列中的任意一个序列的部分或者全部。

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高到低排列的。

在一个可能的设计中，若所述第一序列中的所述n个极化信道的序号是按所述n个极化信道的可靠度从高到低排列的，且所述n个极化信道的序号最小值为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，若所述第一序列中的所述n个极化信道的序号是按所述n个极化信道的可靠度从低到高排列的，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，所述第一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(最左面的记为1)则该信道的可靠度可以表示为n或者归一化的n/n，其中n为序列的长度。

在一个可能的设计中，所述第一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，根据可靠度计算公式确定极化信道的可靠度，获得可靠度排序，按照可靠度排序选择信息比特序号，按照所选择的信息比特序号进行polar码编码，其中，可靠度计算公式中包括可改变的变换核(可以简称为核)，该变换核用于将信号从极化信道序号域转换到可靠度权重域。本申请实施例由于采用包含可改变的变换核的可靠度计算公式，能够同时兼顾极化信道序号域和可靠度权重域的分辨能力，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。

本申请实施例提供的方案涉及如何确定极化信道可靠度。本申请实施例的基本发明思路是：极化信道可靠度可以通过可靠度来表征，从信号谱分析的角度来看，现有的可靠度对极化信道可靠度的近似可以理解为一种信号的域变换。类似于傅里叶变换采用e^jw这个核来实现信号时域与频域间的转换，该方法通过采用β核来将信号从信道序号域转换到可靠度权重域。在信号的时-频分析领域里，最常见的包括傅里叶变换及小波变换。傅里叶变换因受限于三角函数核e^iw的形式，其对信号时-频分析过程中，存在时域分辨力与频域分辨力不可兼得的窘境。小波变化通过采用小波变换核、函数形式多样，因此能够在进行域变换时扑捉到信号在时域上瞬间的变化，能够实现时-频域上同时兼顾的分辨能力。本申请实施例中，通过采用可改变的变换核来对极化信道可靠度进行估计，从而改善序列可靠度估计的精度。

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于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、获取用于对k个待编码比特进行编码的第一序列。

其中，第一序列中包含n个极化信道的序号，n个极化信道的序号在第一序列中是按照n个极化信道的可靠度进行排列的，k为正整数，n为polar码的母码长度，n为2的正整数次幂。

步骤202、根据可靠度从高到低的顺序，从第一序列中选择k个极化信道的序号。

步骤203、按照所选择的k个极化信道的序号放置所述待编码比特，并对待编码比特进行polar码编码。

其中，将所述k个待编码比特映射到所述n个极化信道中的k个极化信道上，其中，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。

可选的，第一序列为第二序列的全部或者子集，第二序列中包含nmax个极化信道的序号，nmax个极化信道的序号在第二序列中是按照nmax个极化信道的可靠度进行排列的，nmax为2的正整数次幂，nmax≥n。nmax个极化信道的可靠度的计算方式与n个极化信道的可靠度的计算方式类似。

可选的，根据目标码长对polar码编码后的序列进行速率匹配。

本实施例提供的编码方法，通过接收到输入的信息比特后，根据polar码的目标码长n确定待编码比特的个数k，无论在线计算还是预先计算并存车的方式，若已知第二序列，则可以从第二序列中获取第一序列。其中，第二序列中包含通信系统所支持的最大码长nmax个极化信道的可靠度排序。可选的，可以从预先存储的第二序列中获取第一序列，接着根据第一序列确定出信息比特，最后对k个待编码比特进行polar编码，获得polar编码后的比特序列。从而实现了通过半在线计算半离线存储的方式获得polar码极化信道的可靠度来确定信息比特与固定比特的位置。

下面具体介绍一下如何确定n个极化信道中第i个极化信道的可靠度。n个极化信道的序号可以为0～(n-1)，也可以为1～n。本申请实施例中，在确定n个极化信道的第i个极化信道的可靠度时，i的取值可以为1、2、…、n，也可以为0、1、…、n-1。

用wi表示n个极化信道中第i个极化信道的可靠度，可靠度计算公式可以包括但不限于以下几种。可以理解的是，本申请实施例中涉及的公式只是一种举例，本领域技术人员可以在对公式进行简单的变形而不影响公式的性能的基础上获得的方案，均属于本申请实施例保护的范围。

以下公式中，不失一般性1≤i≤n，wi为第i个极化信道的可靠度，n＝log2n，i满足i-1＝bn-1bn-2…b0，其中bn-1bn-2...b1b0为i-1的二进制表示，bj为信道序号i的二进制表示中第j位的取值,bj∈{0,1}，这样的取法在于n位二进制数只能表示从0到n-1的数，因此需要将极化信道序号依次减1。当然，极化信道序号i的取值范围也可以选为0≤i≤n-1，此时i＝bn-1bn-2...b1b0，即bn-1bn-2...b1b0为i的二进制表示，其他性质及各极化信道的先后关系均保持不变。因此本申请中，还是以1≤i≤n为例说明。

计算方式一、

公式(1)采用ψ(ξ，j)作为变换核，简称为ψ核。也可以认为ψ(ξ，j)是一种核函数，核函数也可以称为基函数，ψ(ξ，j)的具体形式是阶数(也可以称为阶项)ξ和j的函数。ξ为构成公式(1)的阶数集合，该集合中包括的元素可以是自然数、整数、有理数、或者无理数；c(ξ)是ξ及j的函数、ξ阶ψ核前的系数，表征了该阶项的权重值，即该阶ψ核对最终可靠度贡献的大小；e(ξ)是ξ的函数，表征了该阶ψ核随j变化的频率；ψ(ξ，j)同时是ξ及j的函数，决定了核函数或基函数的具体表现形式。

例如，ψ(ξ)＝a^ξ+b，其中a、b为控制核形式的参数常量；其中d、f和g为控制频率的参数常量；c(ξ)＝c^ξ。

当a＝-0.22，b＝1.2，c＝0.26，d＝1，f＝0，g＝1且ξ＝{0，1}时，公式(1)可以具体化为公式(2)。

需要注意的是，这里举例中ψ、e和c都包含了与ξ的指数型函数关系，实际中ψ、e和c与ξ的数学函数关系不限于此，因数学函数形式的多样性，这里不一一赘述。

计算方式二、

集合ξ为单元素集合，即ξ＝1或者size(ξ)＝1，则公式(1)可以具体化为公式(3)。

其中c和e可以为常数，这样公式(3)可以进一步简化为公式(4)。

公式(4)强调了核函数随极化信道序号二进制位的变化。

公式(3)和公式(4)中各符号意义参考公式(1)的叙述，重复之处不再赘述。

计算方式三、

可靠度可以用极化权重来表征，传统可靠度计算方式可以通过极化权重的计算公式来确定，例如，一种极化权重的计算公式为：β为变换核，为大于1的常量，例如，(这个定义及举例也适用于下面其他计算方式，因此不再赘述)。计算方式三中变换核采用传统可靠度计算方式中的β核形式，通过采用不同阶数的β来组建可改变的变换核。具体计算公式如公式(5)。

c(ξ)是ξ的函数、ξ阶β核前的系数，表征了该阶项的权重值，即该阶β核对最终可靠度贡献的大小；e(ξ)是ξ的函数、决定了各阶β核的具体形式，表征了该阶β核随j变化的频率。其余为描述符号与公式(1)中相同符号的意义相同，重复之处不再赘述。

计算方式四、

若公式(5)中的函数传统可靠度计算方式仅包含0阶β核，如当采用单个低阶核时，传统可靠度计算方式在进行域变换时其对可靠度变化的分辨能力有限，因此可以引入高阶核来提高可靠度的估计精度。

引入高阶核的一种表征方法如公式(6)所示。

其中各符号含义如公式(5)中的解释，在此不再赘述。

例如，ξ＝{0，1}包含0阶项及1阶项、系数定义为时，公式(6)具体化为公式(7)。

又如，ξ＝{0，1，2}包含0阶项、1阶项及2阶项，系数同样定义为时，公式(6)可以具体化为公式(8)。

计算方式五、

公式(1)中的部分ψ核为β核，具体如公式(9)。

ξ1和ξ2包含构成公式(9)的阶数。对于ξ仅属于ξ1的集合，其核函数为β，对于例如：当ξ＝1时，c(1)＝1，ψ(1)＝β＝2^1/4＝1.1892，e(1)＝1；当ξ＝2时，c(2)＝0.17，ψ(2)＝0.63，e(1)＝1/4，或者c(2)＝0.17，ψ(2)＝0.8909，e(1)＝1；

即

或者，

计算方式六、

其中β为一个正的常量，不失一般性，令k为一个包含信道序号二进制表达位的集合，即元素包含0到n-1，k为集合{0,1，…，n-1}的子集，其集合大小0<＝size(k)<＝n-1，如果m出现在k集合中，那么信道序号0到2^m-1与信道序号2^m到2^m+1-1在序列中的排序就会不一致，k集合中的元素会改变序列的对称特性，其中a、c分别为核函数b、d的权重。

利用上述公式(1)～公式(12)中的任一种公式可以确定n个极化信道中第i个极化信道的可靠度，从而获得n个极化信道中每个极化信道的可靠度，按照可靠度从高到低或者从小到大的顺序，可以确定可靠度排序的序列，该序列可以应用于polar码编码的过程中。

本申请实施例提供如下一些可选的序列的举例。在polar编码过程中，可以通过以上公式来获取序列；也可以预先将获取的序列存储下来，应用存储的序列，例如，用查表的方式获取。本申请构造序列的设计，可以与信道参数无关，因此便于离线存储。使用如下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的公式计算获得，也可以通过其他可能的方法获得，本申请实施例不作限定。

若n＝1024，则：

利用公式(2)获得的序列为序列①，序号从1开始(下面所有给出的序列例子中，序号也均从1开始，不再赘述):

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

利用公式(7)获得的序列为序列②:

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

利用公式(8)获得的序列为序列③:

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

利用公式(10)或(11)获得的序列为序列④:

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

若n＝512，则：

利用公式(2)获得的序列为序列⑤:

[1,2,3,5,9,17,4,33,6,7,65,10,11,18,13,19,129,34,21,35,25,8,37,66,257,67,12,41,69,14,20,49,130,15,73,22,131,36,23,133,81,26,38,27,258,137,39,68,97,42,259,29,145,43,70,261,50,71,16,45,74,51,265,132,161,75,24,53,134,82,273,77,135,83,28,57,193,138,40,98,85,260,30,139,289,146,99,44,31,262,141,89,147,72,101,46,263,321,52,266,162,149,47,76,105,267,54,163,274,153,78,55,269,136,165,84,113,58,194,275,385,79,59,86,195,140,290,169,277,100,87,32,291,61,197,142,90,281,148,177,102,264,143,293,91,322,201,150,103,48,106,323,93,268,297,164,151,209,154,107,325,56,270,166,114,276,155,305,386,80,109,271,167,115,329,60,196,225,170,278,387,157,88,117,292,62,171,198,279,389,337,282,178,63,199,144,294,173,92,121,202,283,179,393,104,295,324,94,203,353,298,285,152,181,210,108,401,95,299,326,205,211,156,306,185,110,327,272,301,168,116,330,226,307,213,388,158,417,111,331,118,227,172,280,159,309,390,338,217,119,333,64,200,229,174,391,122,339,284,313,449,180,394,296,175,123,204,354,233,341,286,395,182,402,96,355,125,300,206,287,183,397,345,212,241,186,403,328,207,357,302,308,187,214,418,112,405,303,332,361,228,215,160,310,419,189,218,409,120,334,230,311,392,421,340,219,369,314,450,335,231,176,124,234,315,342,451,221,396,425,356,126,235,343,288,317,453,184,398,346,242,404,433,127,208,358,237,399,347,243,457,188,406,359,304,362,349,216,245,420,190,407,465,410,363,312,191,422,220,370,249,411,336,365,232,423,371,316,452,222,481,426,413,236,344,223,373,318,427,454,434,128,238,319,455,400,429,348,377,244,458,435,360,239,350,459,246,408,437,466,364,351,247,461,192,250,467,412,441,366,424,372,251,482,469,414,367,224,374,483,253,428,415,473,375,320,456,485,430,378,436,240,431,379,460,489,438,352,381,248,462,439,468,497,442,463,252,443,470,368,484,254,471,416,445,474,376,255,486,475,487,432,380,490,477,382,491,440,498,383,464,493,499,444,472,501,446,256,447,476,505,488,478,492,479,384,494,500,495,502,503,448,506,507,480,509,496,504,508,510,511,512]。

利用公式(7)获得的序列为序列⑥:

[1,2,3,5,9,17,33,4,6,7,65,10,11,18,13,129,19,34,21,35,25,257,37,8,66,67,41,12,69,14,130,49,20,15,73,131,22,133,36,23,81,26,258,38,137,27,259,39,97,68,42,29,145,261,70,43,50,71,265,45,16,74,161,132,51,75,134,273,53,24,82,77,135,193,83,138,57,28,289,260,40,98,139,85,30,146,262,99,141,44,31,89,147,263,321,101,72,266,46,162,149,52,267,47,105,76,163,274,54,153,269,385,78,165,136,275,194,55,113,84,58,79,290,195,277,169,140,86,59,291,197,100,87,142,281,61,32,90,177,148,293,264,322,102,143,201,91,150,323,103,297,268,48,106,164,93,151,209,325,154,270,107,386,166,305,276,56,114,155,271,329,109,387,80,167,225,196,278,115,170,157,60,389,292,279,198,337,171,117,88,282,62,178,294,199,393,173,144,283,202,63,121,92,179,295,353,324,104,298,203,285,401,94,181,152,210,326,299,205,108,95,306,211,327,185,156,301,272,330,417,110,388,168,307,226,213,116,158,331,111,390,227,309,280,338,172,118,159,217,333,391,449,229,200,339,394,119,174,313,284,64,122,180,296,354,395,341,175,233,204,286,123,402,182,355,397,300,287,206,345,125,403,96,183,241,212,357,328,186,302,207,418,405,308,214,187,303,361,332,419,112,228,310,215,409,189,160,218,334,421,392,450,311,230,369,340,120,314,219,335,451,231,425,396,342,176,315,234,221,124,453,356,343,398,235,317,288,346,433,126,404,184,242,358,399,457,237,208,347,127,406,243,359,188,304,362,420,349,407,465,245,216,410,190,363,422,312,370,411,191,249,220,365,336,423,481,452,232,371,426,413,316,222,454,427,373,344,236,318,223,434,455,429,400,458,319,238,377,348,435,128,244,360,459,239,350,437,408,466,246,461,364,351,467,247,441,412,192,250,366,424,482,469,372,414,251,367,483,428,374,415,473,253,224,485,456,375,430,320,378,436,431,489,460,240,379,438,462,381,352,439,497,468,248,442,463,470,443,252,368,484,471,445,416,474,254,486,376,475,255,487,432,490,477,380,491,382,440,498,493,464,383,499,444,501,472,446,447,505,476,256,488,478,492,479,494,384,500,495,502,503,448,506,507,509,480,496,504,508,510,511,512]。

利用公式(8)获得的序列为序列⑦:

[1,2,3,5,9,17,33,4,6,65,7,10,11,18,129,13,19,34,21,35,25,257,37,66,8,67,41,12,69,130,14,49,20,73,131,15,22,133,36,81,23,26,258,137,38,27,259,97,39,68,42,145,29,261,70,43,50,265,71,161,45,74,132,16,51,75,273,134,53,82,24,193,77,135,83,138,57,289,28,260,98,139,40,85,146,30,262,99,141,44,89,321,147,31,263,101,266,72,162,46,149,52,267,105,163,47,76,274,153,54,385,269,165,275,194,78,113,136,55,84,58,290,195,79,277,169,140,86,59,291,197,100,281,87,142,177,61,90,293,322,148,32,264,201,102,143,91,323,150,297,103,268,106,164,209,48,93,325,151,154,386,270,107,305,166,276,114,329,155,56,387,271,225,109,167,196,80,278,115,170,157,389,60,292,337,279,198,171,117,282,88,178,62,393,294,199,173,283,202,121,144,353,179,63,92,295,324,298,203,104,401,285,181,210,94,326,152,299,205,108,306,211,95,327,185,417,301,330,156,388,272,307,226,110,168,213,116,331,158,390,227,111,309,338,280,172,217,118,449,333,159,391,229,339,394,200,313,119,174,284,122,354,180,64,395,296,341,233,175,204,402,286,123,355,182,397,300,345,403,287,206,241,125,357,183,212,96,328,186,418,302,207,405,308,214,361,187,419,303,332,228,112,409,310,215,189,218,421,450,334,160,369,392,311,230,340,314,219,120,451,335,425,231,396,342,315,234,176,221,453,124,356,343,398,235,433,317,346,404,288,242,126,457,358,184,399,237,347,208,406,243,127,359,362,188,420,465,304,349,407,245,410,216,363,190,422,370,411,312,249,481,365,191,220,423,452,336,371,426,232,413,316,222,454,427,373,344,236,434,318,223,455,429,458,377,400,435,319,238,348,244,128,459,360,239,437,466,350,408,246,461,364,467,351,441,247,412,250,482,366,192,424,469,372,414,251,483,367,428,473,374,415,253,485,224,456,375,430,378,436,320,489,431,460,379,240,438,462,497,381,439,468,352,442,248,463,470,443,252,484,368,471,445,474,416,254,486,475,376,255,487,490,432,477,380,491,498,382,440,493,464,499,383,444,501,472,446,505,447,476,256,488,478,492,479,494,500,384,495,502,503,506,448,507,509,480,496,504,508,510,511,512]。

利用公式(10)或(11)生成的序列为序列⑧:

[1,2,3,5,9,17,33,4,6,7,65,10,11,18,13,129,19,34,21,35,25,37,8,257,66,67,41,12,69,14,49,130,20,15,73,131,22,36,133,23,81,26,38,258,137,27,39,97,259,68,42,29,145,43,261,70,50,71,45,16,265,74,161,51,132,75,53,134,24,273,82,77,135,193,83,57,138,28,40,289,98,85,260,139,30,146,99,44,262,141,31,89,147,101,263,72,321,46,266,162,52,149,47,105,267,76,163,54,274,153,269,78,165,55,113,136,385,275,194,84,58,79,195,290,169,59,277,86,140,291,100,197,87,61,142,32,281,90,177,148,293,102,264,143,322,201,91,150,103,323,48,297,106,93,268,164,151,209,154,107,325,270,166,56,305,114,386,276,155,109,271,80,329,167,225,115,387,196,170,60,278,157,117,292,171,389,279,198,88,337,62,282,178,199,294,173,63,121,144,393,283,202,92,179,295,104,353,324,203,298,285,94,181,152,401,210,299,108,326,205,95,211,306,185,156,327,301,110,272,330,168,417,307,226,116,213,388,158,111,331,227,309,118,172,390,280,159,338,217,333,229,119,391,200,449,339,174,64,313,122,394,284,180,296,175,354,233,123,341,395,204,286,182,402,355,125,300,397,287,206,96,345,183,241,403,212,186,357,328,207,302,418,308,187,405,214,303,112,361,332,419,228,215,310,189,160,409,218,334,421,311,230,120,369,392,450,340,219,314,335,231,451,176,425,315,234,124,342,221,396,356,235,453,343,317,126,398,288,346,184,433,242,404,358,237,127,399,208,457,347,243,188,406,359,304,362,349,245,420,407,216,465,190,410,363,422,312,191,370,249,411,220,365,336,423,232,481,371,452,426,316,413,222,373,427,236,454,344,223,318,434,455,429,319,238,128,377,400,458,348,435,244,360,239,459,350,437,246,408,466,364,461,351,247,467,192,441,250,412,366,424,482,372,251,469,414,367,483,374,253,428,415,224,473,485,375,456,430,320,378,436,431,240,489,379,460,438,381,462,352,439,248,497,468,442,463,443,252,470,368,484,471,445,254,416,474,486,376,255,475,487,432,490,380,477,491,382,440,498,493,383,464,499,444,501,472,446,447,256,505,476,488,478,492,479,494,384,500,495,502,503,448,506,507,509,480,496,504,508,510,511,512]。

若n＝256，则：

利用公式(2)获得的序列为序列⑨:

[1,2,3,5,9,17,4,33,6,7,65,10,11,18,13,19,129,34,21,35,25,8,37,66,67,12,41,69,14,20,49,130,15,73,22,131,36,23,133,81,26,38,27,137,39,68,97,42,29,145,43,70,50,71,16,45,74,51,132,161,75,24,53,134,82,77,135,83,28,57,193,138,40,98,85,30,139,146,99,44,31,141,89,147,72,101,46,52,162,149,47,76,105,54,163,153,78,55,136,165,84,113,58,194,79,59,86,195,140,169,100,87,32,61,197,142,90,148,177,102,143,91,201,150,103,48,106,93,164,151,209,154,107,56,166,114,155,80,109,167,115,60,196,225,170,157,88,117,62,171,198,178,63,199,144,173,92,121,202,179,104,94,203,152,181,210,108,95,205,211,156,185,110,168,116,226,213,158,111,118,227,172,159,217,119,64,200,229,174,122,180,175,123,204,233,182,96,125,206,183,212,241,186,207,187,214,112,228,215,160,189,218,120,230,219,231,176,124,234,221,126,235,184,242,127,208,237,243,188,216,245,190,191,220,249,232,222,236,223,128,238,244,239,246,247,192,250,251,224,253,240,248,252,254,255,256]。

利用公式(7)获得的序列为序列⑩:

[1,2,3,5,9,17,33,4,6,7,65,10,11,18,13,129,19,34,21,35,25,37,8,66,67,41,12,69,14,130,49,20,15,73,131,22,133,36,23,81,26,38,137,27,39,97,68,42,29,145,70,43,50,71,45,16,74,161,132,51,75,134,53,24,82,77,135,193,83,138,57,28,40,98,139,85,30,146,99,141,44,31,89,147,101,72,46,162,149,52,47,105,76,163,54,153,78,165,136,194,55,113,84,58,79,195,169,140,86,59,197,100,87,142,61,32,90,177,148,102,143,201,91,150,103,48,106,164,93,151,209,154,107,166,56,114,155,109,80,167,225,196,115,170,157,60,198,171,117,88,62,178,199,173,144,202,63,121,92,179,104,203,94,181,152,210,205,108,95,211,185,156,110,168,226,213,116,158,111,227,172,118,159,217,229,200,119,174,64,122,180,175,233,204,123,182,206,125,96,183,241,212,186,207,214,187,112,228,215,189,160,218,230,120,219,231,176,234,221,124,235,126,184,242,237,208,127,243,188,245,216,190,191,249,220,232,222,236,223,238,128,244,239,246,247,192,250,251,253,224,240,248,252,254,255,256]。

利用公式(8)获得的序列为序列

[1,2,3,5,9,17,33,4,6,65,7,10,11,18,129,13,19,34,21,35,25,37,66,8,67,41,12,69,130,14,49,20,73,131,15,22,133,36,81,23,26,137,38,27,97,39,68,42,145,29,70,43,50,71,161,45,74,132,16,51,75,134,53,82,24,193,77,135,83,138,57,28,98,139,40,85,146,30,99,141,44,89,147,31,101,72,162,46,149,52,105,163,47,76,153,54,165,194,78,113,136,55,84,58,195,79,169,140,86,59,197,100,87,142,177,61,90,148,32,201,102,143,91,150,103,106,164,209,48,93,151,154,107,166,114,155,56,225,109,167,196,80,115,170,157,60,198,171,117,88,178,62,199,173,202,121,144,179,63,92,203,104,181,210,94,152,205,108,211,95,185,156,226,110,168,213,116,158,227,111,172,217,118,159,229,200,119,174,122,180,64,233,175,204,123,182,206,241,125,183,212,96,186,207,214,187,228,112,215,189,218,160,230,219,120,231,234,176,221,124,235,242,126,184,237,208,243,127,188,245,216,190,249,191,220,232,222,236,223,238,244,128,239,246,247,250,192,251,253,224,240,248,252,254,255,256]。

利用公式(10)或(11)生成的序列为序列

[1,2,3,5,9,17,33,4,6,7,65,10,11,18,13,129,19,34,21,35,25,37,8,66,67,41,12,69,14,49,130,20,15,73,131,22,36,133,23,81,26,38,137,27,39,97,68,42,29,145,43,70,50,71,45,16,74,161,51,132,75,53,134,24,82,77,135,193,83,57,138,28,40,98,85,139,30,146,99,44,141,31,89,147,101,72,46,162,52,149,47,105,76,163,54,153,78,165,55,113,136,194,84,58,79,195,169,59,86,140,100,197,87,61,142,32,90,177,148,102,143,201,91,150,103,48,106,93,164,151,209,154,107,166,56,114,155,109,80,167,225,115,196,170,60,157,117,171,198,88,62,178,199,173,63,121,144,202,92,179,104,203,94,181,152,210,108,205,95,211,185,156,110,168,226,116,213,158,111,227,118,172,159,217,229,119,200,174,64,122,180,175,233,123,204,182,125,206,96,183,241,212,186,207,187,214,112,228,215,189,160,218,230,120,219,231,176,234,124,221,235,126,184,242,237,127,208,243,188,245,216,190,191,249,220,232,222,236,223,238,128,244,239,246,247,192,250,251,253,224,240,248,252,254,255,256]。

若n＝128，则：

利用公式(2)获得的序列为序列

[1,2,3,5,9,17,4,33,6,7,65,10,11,18,13,19,34,21,35,25,8,37,66,67,12,41,69,14,20,49,15,73,22,36,23,81,26,38,27,39,68,97,42,29,43,70,50,71,16,45,74,51,75,24,53,82,77,83,28,57,40,98,85,30,99,44,31,89,72,101,46,52,47,76,105,54,78,55,84,113,58,79,59,86,100,87,32,61,90,102,91,103,48,106,93,107,56,114,80,109,115,60,88,117,62,63,92,121,104,94,108,95,110,116,111,118,119,64,122,123,96,125,112,120,124,126,127,128]。

利用公式(7)获得的序列为序列

[1,2,3,5,9,17,33,4,6,7,65,10,11,18,13,19,34,21,35,25,37,8,66,67,41,12,69,14,49,20,15,73,22,36,23,81,26,38,27,39,97,68,42,29,70,43,50,71,45,16,74,51,75,53,24,82,77,83,57,28,40,98,85,30,99,44,31,89,101,72,46,52,47,105,76,54,78,55,113,84,58,79,86,59,100,87,61,32,90,102,91,103,48,106,93,107,56,114,109,80,115,60,117,88,62,63,121,92,104,94,108,95,110,116,111,118,119,64,122,123,125,96,112,120,124,126,127,128]。

利用公式(8)获得的序列为序列

[1,2,3,5,9,17,33,4,6,65,7,10,11,18,13,19,34,21,35,25,37,66,8,67,41,12,69,14,49,20,73,15,22,36,81,23,26,38,27,97,39,68,42,29,70,43,50,71,45,74,16,51,75,53,82,24,77,83,57,28,98,40,85,30,99,44,89,31,101,72,46,52,105,47,76,54,78,113,55,84,58,79,86,59,100,87,61,90,32,102,91,103,106,48,93,107,114,56,109,80,115,60,117,88,62,121,63,92,104,94,108,95,110,116,111,118,119,122,64,123,125,96,112,120,124,126,127,128]。

利用公式(10)或(11)获得的序列为序列

[1,2,3,5,9,17,33,4,6,7,65,10,11,18,13,19,34,21,35,25,37,8,66,67,41,12,69,14,49,20,15,73,22,36,23,81,26,38,27,39,97,68,42,29,43,70,50,71,45,16,74,51,75,53,24,82,77,83,57,28,40,98,85,30,99,44,31,89,101,72,46,52,47,105,76,54,78,55,113,84,58,79,59,86,100,87,61,32,90,102,91,103,48,106,93,107,56,114,109,80,115,60,117,88,62,63,121,92,104,94,108,95,110,116,111,118,119,64,122,123,125,96,112,120,124,126,127,128]。

若n＝64，则：

利用公式(2)获得的序列为序列

[1,2,3,5,9,17,4,33,6,7,10,11,18,13,19,34,21,35,25,8,37,12,41,14,20,49,15,22,36,23,26,38,27,39,42,29,43,50,16,45,51,24,53,28,57,40,30,44,31,46,52,47,54,55,58,59,32,61,48,56,60,62,63,64]。

利用公式(7)获得成的序列为序列

[1,2,3,5,9,17,33,4,6,7,10,11,18,13,19,34,21,35,25,37,8,41,12,14,49,20,15,22,36,23,26,38,27,39,42,29,43,50,45,16,51,53,24,57,28,40,30,44,31,46,52,47,54,55,58,59,61,32,48,56,60,62,63,64]。

利用公式(8)获得的序列为序列

[1,2,3,5,9,17,33,4,6,7,10,11,18,13,19,34,21,35,25,37,8,41,12,14,49,20,15,22,36,23,26,38,27,39,42,29,43,50,45,16,51,53,24,57,28,40,30,44,31,46,52,47,54,55,58,59,61,32,48,56,60,62,63,64]。

利用公式(10)或(11)获得的序列为序列

[1,2,3,5,9,17,33,4,6,7,10,11,18,13,19,34,21,35,25,37,8,41,12,14,49,20,15,22,36,23,26,38,27,39,42,29,43,50,45,16,51,53,24,57,28,40,30,44,31,46,52,47,54,55,58,59,61,32,48,56,60,62,63,64]。

需要说明的是，上述一些序列只是一些举例，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、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、序列中的部分元素进行调整，但是根据该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、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序列中所表示的极化信道的具体位置。

4、上述序列中的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，将上述序列中的元素逆序排列或者反序排列即可。

5、上述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(最左面的记为1)则该信道的可靠度可以表示为n或者归一化的n/n，其中n为序列的长度。

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：

获取单元301，用于获取用于对k个待编码比特进行编码的第一序列，第一序列中包含n个极化信道的序号，n个极化信道的序号在第一序列中是按照n个极化信道的可靠度进行排列的，k为正整数，n为polar码的母码长度，n为2的正整数次幂，k≤n；

选择单元302，用于根据可靠度从高到低的顺序，从第一序列中选择k个极化信道的序号；

编码单元303，用于按照所选择的k个极化信道的序号放置待编码比特，并对待编码比特进行polar码编码。

其中，第一序列可以为上述任一种举例的序列，n个极化信道中第i个极化信道的可靠度也可以通过上述任一种举例的公式来确定。

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。