CN108667568A

CN108667568A - 一种Polar码编码方法及装置

Info

Publication number: CN108667568A
Application number: CN201810709840.3A
Authority: CN
Inventors: 皇甫幼睿; 王坚; 李榕; 乔云飞; 王俊
Original assignee: Huawei Technologies Co Ltd
Current assignee: Huawei Technologies Co Ltd
Priority date: 2017-07-28
Filing date: 2017-07-28
Publication date: 2018-10-16
Anticipated expiration: 2037-07-28
Also published as: CN109309503B; JP2020528250A; ES2923915T3; CN109309503A; KR102287176B1; RU2020108334A; EP3584937B1; US20210184695A1; CN111030707A; BR112020001338A2; EP3584937A4; WO2019019852A1; RU2020108334A3; EP4084375A1; RU2760317C2; EP3584937A1; US10917115B2; CN111030707B; BR112020001338B1; KR20200015779A

Abstract

本申请涉及通信技术领域，公开了一种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。该方法为：确定用于对待编码比特进行编码的序列，采用所述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，所述序列用于表征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。

Description

一种Polar码编码方法及装置

技术领域

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(low densityparity check，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(5th generation，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(enhance mobile broadband，eMBB)、海量机器连接通信(massive machine typecommunication，mMTC)和高可靠低延迟通信(ultra reliable low latencycommunication，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。

发明内容

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，根据性能优劣判断规则选取性能较好的至少两个序列，确定该至少两个序列中位于相同排序位置的相同序号，根据将所确定的该相同序号确定为序列中可靠度不变的序号，且可靠度不变的序号在序列中的排序按照该至少两个序列中的排序顺序，对比该至少两个序列中除已确定的可靠度不变的序号之外的其余序号，选择出一些子序列，子序列中的元素位置可以相互交换。由此获得的偏序序列作为本申请实施例中用于Polar码编码的排序序列。这样，通过上述方法获得的排序序列能够综合考量多种性能较好的序列，通过不同序列之间的交集确定可靠度不变的序号，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序列，所述序列用于表征N个极化信道的可靠度排序，采用所述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，根据所述母码长度N和预设的序列确定所述用于对待编码比特进行编码的序列。

在一个可能的设计中，所述用于对待编码比特进行编码的序列为另一序列的全部或者子集，所述另一序列中包含N_max个序号，所述N_max个序号在所述另一序列中是按照所述N_max个极化信道的可靠度进行排列的，所述N_max为2的正整数次幂，所述N_max≥N。

在一个可能的设计中，所述序列符合嵌套特性，所述嵌套特性为：根据较大母码长度N₁的序列获得较短母码长度N₂的序列，从较大母码长度N₁的序列中取出序号小于等于N₂的元素，所得子序列即为较短母码长度N₂的序列，其中，所获得的较短母码长度N₂的序列中元素的排序与这些元素在较大母码长度N₁的序列中的排序相同。

在一个可能的设计中，所述N_max为Polar码的最大母码长度，例如，所述N_max＝1024，或者，所述N_max＝512。序列要符合嵌套特性，当最大母码长度的序列确定，可以根据最大母码长度的序列获得其它较短母码长度的序列。例如，确定最大母码长度为1024的长度为1024的序列，将长度为1024的序列中元素值小于等于64的元素取出，组成长度为64的子序列，所取出的长度为64的子序列中的元素排序与这些元素在长度为1024的序列中的排序相同，即符合嵌套特性。类似的，可以从长度为1024的序列中取出长度为128、256和512的序列。将取出的这些短码母码长度的序列应用到本申请实施例中以用于对待编码比特进行编码。

在一个可能的设计中，所述序列可以为说明书中序列1)～30)中的任意一个序列的部分或者全部；在N＝1024时，所述序列可以为说明书中序列S1～S20中的任意一个序列的部分或者全部。

在一个可能的设计中，当N取值64、128、256或512时，所述序列可以根据嵌套特性从预设的最大母码长度的序列中取出，也可以直接应用说明书中序列1)～30)中对应码长的序列。

通过所述序列进行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S1～S20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，则将说明书中序列S1～S20中的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(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，说明书中序列S1～S20中的任意一个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，根据性能优劣判断规则选取性能较好的至少两个序列，确定该至少两个序列中位于相同排序位置的相同序号，根据将所确定的该相同序号确定为序列中可靠度不变的序号，且可靠度不变的序号在序列中的排序按照该至少两个序列中的排序顺序，对比该至少两个序列中除已确定的可靠度不变的序号之外的其余序号，选择出一些子序列，子序列中的元素位置可以相互交换。由此获得的偏序序列作为本申请实施例中用于Polar码编码的排序序列。这样，通过上述方法获得的排序序列能够综合考量多种性能较好的序列，通过不同序列之间的交集确定可靠度不变的序号，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G_N，编码过程为其中是一个二进制的行矢量，长度为N(即码长)；G_N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(frozen bits)，其索引的集合用的补集表示。Polar码的编码过程相当于：这里，G_N(A)是G_N中由集合中的索引对应的那些行得到的子矩阵，G_N(AC)是G_N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G_N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，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，将这种序列称为Q序列。

另一种情况下，该序列中可以包括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，信道x在序列的排序位置为y，序列中最低序号记为1，则该信道的可靠度可以表示为y或者归一化的y/N。类似的，N个极化信道的可靠度的值在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。

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，以序列中元素取值为1～N为例进行说明，可知，将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的序列，两种序列的性能相同，均可表征N个极化信道可靠度排序。本申请实施例涉及的序列，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序列，两种序列的性能相同，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(long termevolution，LTE)系统中，或者，第五代(5th generation，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(mobile station，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序列。

其中，该序列用于表征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。

具体地，根据确定的用于对待编码比特进行编码的序列，按照可靠度由高到低的顺序，选择K个极化信道的序号，将K个待编码比特放置在所选择的K个极化信道。

实际应用中，可将上述用于对待编码比特进行编码的序列离线存储，给定传输参数(例如码长和码率)后，利用预先存储的序列来确定构造Polar码所用的信息比特集合和冻结比特集合，具体确定方法可以参照上述对Polar码编码过程的介绍，在此不再赘述。另外，本申请实施例中，若上述用于对待编码比特进行编码的序列为较短码长的序列，还可以根据序列的嵌套特性，从存储的较长母码长度的序列中取出较短码长的序列。

下面具体给出一些各种母码长度下的Q序列和Z序列的举例，所给出的Q序列和Z序列均可以用于对待编码比特进行编码。以下举例中，序列中的元素是按照N个极化信道的可靠度排序由低到高的顺序进行排列的。用为表示序列中的子序列，n为子序列在序列的序号，1≤n≤N，n为正整数，m表示序列中第n个子序列的长度，子序列中的元素位置在中是可变的，序列中的子序列外的部分位置的序号是固定的。当子序列中的元素位置在中发生调换后，可以根据以下给出的序列举例获得调换后的序列。

例一、

1)、在N＝64时，序列为Q序列，Q序列中可以包括11个元素位置可变的子序列。n取值1～11，序列可以为：

一种可能的示例中，可以为：

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

2)在N＝64时，序列为Z序列，Z序列中可以包括2个元素位置可变的子序列。n取值1～2。序列可以为

一种可能的示例中，可以为：

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

3)在N＝128时，序列为Q序列，Q序列中可以包括23个元素位置可变的子序列。n取值1～23。序列可以为：

一种可能的示例中，可以为：

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

4)在N＝128时，序列为Z序列，Z序列中可以包括一个元素位置可变的子序列。n取值为1。序列可以为：

一种可能的示例中，可以为：

在这种示例的情况下，Z序列具体化为：

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

5)在N＝256时，序列为Q序列，Q序列中可以包括25个元素位置可变的子序列，n取值1～25。序列可以为：

一种可能的示例中，可以为：

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

6)在N＝256时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

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

7)在N＝512时，序列为Q序列，Q序列中可以包括10个元素位置可变的子序列，n取值1～10。序列可以为：

一种可能的示例中，可以为：

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

8)、在N＝512时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

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

9)、在N＝1024时，序列为Q序列，Q序列中可以包括3个元素位置可变的子序列，n取值1～3。序列可以为：

一种可能的示例中，可以为：

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

10)在N＝1024时，序列为Z序列，Z序列中可以包括1个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

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

例二、

11)、在N＝64时，序列为Q序列，Q序列中可以包括12个元素位置可变的子序列，n取值1～12。序列可以为：

一种可能的示例中，可以为：

在这种示例的情况下，Q序列具体化为[1,2,(5,9,3),17,(33,7,4,13,6,19,10,34,18,11),21,35,25,(8,37),(12,41),(20,14,49),15,22,(27,36,39,23,38,26),(42,29),(50,43,45,16,51),(24,53),(57,28),40,30,(44,31),46,52,47,54,55,58,59,(61,32),48,56,60,62,63,64]

12)、在N＝64时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

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

13)、在N＝128时，序列为Q序列，Q序列中可以包括23(n＝23)个元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

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

14)、在N＝128时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

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

15)、在N＝256时，序列为Q序列，Q序列中可以包括16个元素位置可变的子序列，n取值1～16。序列可以为：

一种可能的示例中，可以为：

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

16)、在N＝256时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

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

17)在N＝512时，序列为Q序列，Q序列中可以包括6个元素位置可变的子序列，n取值1～6。。序列可以为：

一种可能的示例中，可以为：

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

18)在N＝512时，序列为Z序列，Z序列中可以包括1个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

19)在N＝1024时，序列为Q序列，Q序列中可以包括3个元素位置可变的子序列，n取值1～3。。序列可以为：

一种可能的示例中，可以为：

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

20)在N＝1024时，序列为Z序列，Z序列中可以包括1个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

例三、

21)、在N＝64时，序列为Q序列，Q序列中可以包括11个元素位置可变的子序列，n取值1～11。。序列可以为：

一种可能的示例中，可以为：

N＝64，上述Q序列中子序列中的元素位置可以在子序列中发生调换，例如可以为(5,9,3)，也可以调换为(3,9,5)，或者(5,3,9)等等。

在这种示例的情况下，Q序列具体化为[1,2,(5,9,3),17,(33,7,4,13,6,19,10,34,18,11,21,35,25,8,37),(12,41),(20,14,49),15,22,(27,36,39,23,38,26),(42,29),(50,43,45,16,51),(24,53),(57,28,40),30,(44,31),46,52,47,54,55,58,59,(61,32),48,56,60,62,63,64]，其中圆括号中的集合为上述子序列

22)在N＝64时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

中的元素位置可以在子序列中发生调换。

23)在N＝128时，序列为Q序列，Q序列中可以包括17个元素位置可变的子序列，n取值1～17。序列可以为：

一种可能的示例中，可以为：

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

34)在N＝128时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

在这种示例的情况下，Z序列具体化为：

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

25)在N＝256时，序列为Q序列，Q序列中可以包括9个元素位置可变的子序列，n取值1～9。序列可以为：

一种可能的示例中，可以为：

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

26)在N＝256时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

27)在N＝512时，序列为Q序列，Q序列中可以包括4个元素位置可变的子序列，n取值1～4。序列可以为：

一种可能的示例中，可以为：

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

28)在N＝512时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

29)在N＝1024时，序列为Q序列，Q序列中可以包括3个(n＝3)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

30)在N＝1024时，序列为Z序列，Z序列中可以包括一个(n＝1)元素位置可变的子序列。序列可以为：

一种可能的示例中，可以为：

以上举例的序列中，子序列中的元素位置可以在子序列中进行调整，例如，例一的第1)个举例中，那么序列中的第1个子序列中的元素位置可以调整为(5,3,9)，或者(3,5,9)或者(3,9,5)或者(5,9,3)或者(9,3,5)或者(9,5,3)等，调整后获得新序列。以上具体化后的序列中，圆括号中的元素集合表示上述子序列圆括号内的元素位置可以调换，圆括号内的元素位置与圆括号外的元素位置不可调换。

下面对本申请实施例中根据序列进行Polar码编码的过程进行举例说明。

本申请实施例中的序列满足简单的嵌套特征，即可以根据较大母码长度的序列获得其它较短母码长度的序列。具体地，例如根据较大母码长度N₁的序列获得较短母码长度N₂的序列，从较大母码长度N₁的序列中取出序号小于等于N₂的元素，所得子序列即为较短母码长度N₂的序列，其中，所获得的较短母码长度N₂的序列中元素的排序与这些元素在较大母码长度N₁的序列中的排序相同。若长度为K的待编码比特包括：长度为K_info的信息比特和长度为K_CRC的校验比特，编码长度为M，母码长度N，K＝27,K_info＝8，K_CRC＝19，M＝60，N＝64，在构造Polar的参数时，需要获取N＝64的序列，假设获取的序列为Q序列。本申请实施例可以直接使用上述举例序列中N＝64的Q序列，也可以从N>64的Q序列中根据嵌套特性得到N＝64的Q序列。例如利用例一中N＝128的Q序列，取其中子信道序号小于等于64的所有序号，且保持取得的序号的顺序不变，组合成一个新的N＝64的序列:[1,2,(5,9,3),17,33,(7,4,13,6,19,10,34,18,11),21,35,25,(8,37),(12,41),(20,14,49),15,22,(27,36,39,23,38,26),42,29,(50,43,45,16,51,24,53),(57,28,40),30,(44,31,46),52,47,54,(55,58),59,61,32,48,56,60,62,63,64]。由于序列中部分子序列内的元素位置可调，根据嵌套特性由N＝128的序列中获得的新的N＝64的序列和例一的1)中N＝64序列稍有不同，圆括号范围更大，能够组合出来的序列更多。

以直接取得例一中第1)种N＝64的Q序列为例，根据获得的Q序列，按照可靠度由高到低的顺序选择K个极化信道的位置放置待编码比特，K＝27。即在极化信道序号为(43,45,16,51,24,53,57,28,40,30,44,31,46,52,47,54,55,58,59,61,32,48,56,60,62,63,64)的位置或者在极化信道序号为(50,45,16,51,24,53,57,28,40,30,44,31,46,52,47,54,55,58,59,61,32,48,56,60,62,63,64)的位置上放置待编码比特。其中，直接取得的N＝64的Q序列中，在子序列位置上两个元素位置(50,43)可互换，所以有两种可能的组合，由于这些子序列对应的极化信道都会放置待编码比特，地位等价，不用考虑是否调换的问题。

上述放置信息比特的举例的前提条件是，该27个待编码比特的发送优先级相同。如果待编码比特的优先级不同，例如需要将18个校验比特放在选出的27个极化信道中可靠度最高的18个极化信道的位置，1个校验比特要放置在可靠度小于某些信息比特的位置，那么这个校验比特在放置时可能会牵涉到在到这4个子序列中的顺序问题，例如这个校验比特要放置在的第一个位置，当时，第一个位置是16，当16和51的顺序调换后，第一个位置是51，因此将这个校验比特是选择16还是51对最终的性能会产生影响。

除放置待编码比特的其它位置为冻结比特位置。根据待编码比特的位置和冻结比特的位置进行Polar编码和速率匹配。

还可以根据嵌套特性，通过N＝256的序列或者N＝512的序列或N＝1024的序列，获得N＝64的新序列。

以上是以N＝64举例进行说明，其他码长的编码方法类似，在此不再赘述。

本申请实施例上述举例的1)～30)序列可以通过若干组序列取交集获得，例如，通过对若干组母码码长为64的序列取交集获得上述举例的第1)种序列，当然也可以通过其他方式获得上述序列。

一种可能的示例中，通过仿真获得以下序列S1～序列S20，序列S1～序列S20为母码长度为1024的序列，根据嵌套特性，可以分别根据S1～序列S20获得母码长度为64、128、256、512的序列，由此获得20组母码码长分别为64、128、256、512和1024的序列，可以根据第1～5组序列，获得上述例一的1)～10)序列，根据第1～10组序列获得上述例二的11)～20)序列，根据第1～20组序列获得上述例三的21)～30)序列。可以理解，组合的序列数越多，所得序列中位置不变的元素越少，位置可变元素的越多，所得序列包含的范围越大。

本申请实施例还可以采用序列S1～序列S20中的任一种进行图2所示Polar码编码方法。

具体地，序列S1～序列S20可以为：

序列S1：

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

序列S2：

[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]。

序列S3：

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

序列S4：

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

序列S5：

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

序列S6：

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

序列S7：

[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]。

序列S8：

[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]。

序列S9：

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

序列S10：

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

序列S11：

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

序列S12：

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

序列S13：

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

序列S14：

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

序列S15：

[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]。

序列S16：

[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]。

序列S17：

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

序列S18：

[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]。

序列S19：

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

序列S20：

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

以上仿真获得的性能较佳的序列可以通过预设的性能优劣判断规则进行比较，可选的，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。

一种可能的性能优劣判断规如下所述。

评估度量：误块率(block error rate，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WinCount A＝∑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>WinCount Wx,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。

需要说明的是，上述序列S1～S20可以应用到本申请实施例中图2所示的polar码编码方法中，其中任一种序列中，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y(排序序列最小序号记为1)，则该信道的可靠度可以表示为y或者归一化的y/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序列，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，该序列可以是上述方法中所述的序列1)～30)中的任一种序列的部分或全部，也可以是上述方法中序列S1～S20中的任一种序列的部分或全部。

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(central processing unit，CPU)，网络处理器(network processor，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-specific integrated circuit，ASIC)，可编程逻辑器件(programmablelogic device，PLD)或其组合。上述PLD可以是复杂可编程逻辑器件(complexprogrammable logic device，CPLD)，现场可编程逻辑门阵列(field-programmable gatearray，FPGA)，通用阵列逻辑(generic array logic,GAL)或其任意组合。

存储器501可以包括易失性存储器(volatile memory)，例如随机存取存储器(random-access memory，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(flash memory)，硬盘(hard disk drive，HDD)或固态硬盘(solid-state drive，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。

Claims

1.一种极化Polar码编码方法，其特征在于，包括：

确定用于对待编码比特进行编码的序列，所述序列用于表征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；所述用于对待编码比特进行编码的序列为另一序列的全部或者子集，所述另一序列中包含N_max个序号，所述N_max个序号在所述另一序列中是按照所述N_max个极化信道的可靠度进行排列的，所述N_max为2的正整数次幂，所述N_max≥N；

采用所述序列对所述待编码比特进行Polar码编码，得到编码后的比特。

2.如权利要求1所述的方法，其特征在于，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。

3.如权利要求1或2所述的方法，其特征在于，N_max＝1024。

4.如权利要求3所述的方法，其特征在于，所述另一序列为：

其中，为所述序列中的子序列，n为所述子序列在所述另一序列中的序号，n∈{1、2、3}，m表示第n个所述子序列的长度，所述子序列中的元素位置在所述子序列中可以相互交换。

5.如权利要求4所述的方法，其特征在于，所述满足以下至少一种条件：

6.一种极化Polar码编码装置，其特征在于，包括：

确定单元，用于确定用于对待编码比特进行编码的序列，所述序列用于表征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；所述用于对待编码比特进行编码的序列为另一序列的全部或者子集，所述另一序列中包含N_max个序号，所述N_max个序号在所述另一序列中是按照所述N_max个极化信道的可靠度进行排列的，所述N_max为2的正整数次幂，所述N_max≥N；

编码单元，用于采用所述序列对所述待编码比特进行Polar码编码，得到编码后的比特。

7.如权利要求6所述的装置，其特征在于，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。

8.如权利要求5或6所述的装置，其特征在于，N_max＝1024。

9.如权利要求8所述的装置，其特征在于，所述另一序列为：

其中，为所述另一序列中的子序列，n为所述子序列在所述另一序列中的序号，n∈{1、2、3}，m表示第n个所述子序列的长度，所述子序列中的元素位置在所述子序列中可以相互交换。

10.如权利要求9所述的装置，其特征在于，所述满足以下至少一种条件：

11.一种极化Polar码编码装置，其特征在于，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，所述处理器用于执行如下方法：

12.如权利要求11所述的装置，其特征在于，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。

13.如权利要求11或12所述的装置，其特征在于，N_max＝1024。

14.如权利要求13所述的装置，其特征在于，所述另一序列为：

15.如权利要求14所述的装置，其特征在于，所述满足以下至少一种条件：

16.一种Polar码编码装置，其特征在于，包括：

输入接口电路，用于获取待编码比特；

逻辑电路，用于基于获取的待编码比特执行如下方法：确定用于对待编码比特进行编码的序列，所述序列用于表征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；采用所述序列对所述待编码比特进行Polar码编码，得到编码后的比特；得到基于Polar码编码后的比特；所述用于对待编码比特进行编码的序列为另一序列的全部或者子集，所述另一序列中包含N_max个序号，所述N_max个序号在所述另一序列中是按照所述N_max个极化信道的可靠度进行排列的，所述N_max为2的正整数次幂，所述N_max≥N；

输出接口电路，用于输出Polar码编码后的比特。

17.如权利要求16所述的装置，其特征在于，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。

18.如权利要求16或17所述的装置，其特征在于，N_max＝1024。

19.如权利要求18所述的装置，其特征在于，所述另一序列为：

20.如权利要求19所述的装置，其特征在于，所述满足以下至少一种条件：