CN109286403A - 极化码编码的方法和装置 - Google Patents
极化码编码的方法和装置 Download PDFInfo
- Publication number
- CN109286403A CN109286403A CN201710602871.4A CN201710602871A CN109286403A CN 109286403 A CN109286403 A CN 109286403A CN 201710602871 A CN201710602871 A CN 201710602871A CN 109286403 A CN109286403 A CN 109286403A
- Authority
- CN
- China
- Prior art keywords
- sequence
- index
- length
- polarisation channel
- polarization
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
Classifications
-
- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
- H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
- H03M13/13—Linear codes
-
- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L1/00—Arrangements for detecting or preventing errors in the information received
- H04L1/0001—Systems modifying transmission characteristics according to link quality, e.g. power backoff
- H04L1/0009—Systems modifying transmission characteristics according to link quality, e.g. power backoff by adapting the channel coding
-
- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L1/00—Arrangements for detecting or preventing errors in the information received
- H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
- H04L1/0056—Systems characterized by the type of code used
- H04L1/0057—Block codes
Landscapes
- Engineering & Computer Science (AREA)
- Computer Networks & Wireless Communication (AREA)
- Signal Processing (AREA)
- Quality & Reliability (AREA)
- Physics & Mathematics (AREA)
- Probability & Statistics with Applications (AREA)
- Theoretical Computer Science (AREA)
- Error Detection And Correction (AREA)
- Compression, Expansion, Code Conversion, And Decoders (AREA)
Abstract
本申请提供了一种极化码编码的方法和装置,能够提高极化码的性能。该方法包括:发送设备获取所需母码长度对应的排序序列;发送设备获取待编码比特;发送设备采用该所需母码长度对应的排序序列对待编码比特进行极化编码,得到编码后的序列。
Description
技术领域
本申请涉及信道编码领域,并且更具体地,涉及一种极化码编码的方法和装置。
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。极化(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,未来的通信系统(例如,5G)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 MobileBroadband,eMBB)、海量机器连接通信(Massive Machine Type Communication,mMTC)和高可靠低延迟通信(Ultra Reliable Low Latency Communication,URLLC)。这些通信场景对于极化码的编译码性能提出了更高的要求。
而现阶段极化码在应用过程中的性能还不理想,需要进一步改善和提高。
发明内容
本申请提供一种极化码编码的方法和装置,能够提高极化码的性能。
第一方面,本申请提供一种极化码编码的方法,该方法包括:发送设备获取所需母码长度对应的排序序列;发送设备获取待编码比特;发送设备采用该所需母码长度对应的排序序列对待编码比特进行极化编码,得到编码后的序列。
这里的“所需母码长度对应的排序序列”是指,在极化编码时根据信息比特数目K实际使用等于母码长度的排序序列。
在一种可能的实现方式中,发送设备获取所需母码长度对应的排序序列,包括:发送设备从长度为N0的序列中读取所述所需母码长度对应的排序序列,该长度为N0的序列是最大母码长度对应的排序序列或可靠度归一化序列,其中,该最大母码长度对应的排序序列包括N0个极化信道的索引,该N0个极化信道的索引是按照该N0个极化信道的可靠度的大小排序的,该可靠度归一化序列包括N0个极化信道的索引,每个索引对应一个极化信道的可靠度的量化值,N0大于0且为整数。
计算N0个极化信道的可靠度,将这N0个极化信道的可靠度进行量化(或者说,归一化),得到每个极化信道的可靠度的量化值。例如N0=1024,将这N0个量化值中的最大量化值赋值1023,再将次大的量化值赋值1022,依次类推,直至将最小的量化值赋值0。又例如,N0=128,将这N0个量化值中的最大量化值赋值127,再将次大的量化值赋值126,依次类推,直至将最小的量化值赋值0。
在一种可能的实现方式中,发送设备采用该排序序列对待编码比特进行极化编码,得到编码后的序列,包括:发送设备根据该排序序列,确定信息比特索引集合;发送设备根据信息比特索引集合,对待编码比特进行极化编码,得到编码后的序列。
第二方面,本申请提供一种译码方法,该方法包括:接收设备获取所需母码长度对应的排序序列和待译码序列;接收设备采用该所需母码长度对应的排序序列对待译码序列进行译码,得到译码后的序列。
在一种可能的实现方式中,该长度为N0的序列如序列1至序列32中的任意一个。
这里的所说的序列1至序列32详见说明书。
需要说明的是,说明书中的序列1至32中所示的长度为N0的序列的索引均是从0开始的。如果该长度为N0的序列的索引从1开始,则该长度为N0的序列和序列1至序列32中任意一个所示的序列仅仅是形式上的不同,其实质所表示的该N0个极化信道的可靠度的排序是一致的,根据序列1至序列32可以推到出索引从1开始的长度为N0的序列。为了避免繁琐,本文中不再给出索引从1开始的该长度为N0的序列的具体形式。
在一种可能的实现方式中,该最大母码长度对应的排序序列是通过将最大母码长度对应的初始排序序列中小于N的索引依次替换为第一序列包括的索引得到的,
其中,该最大母码长度对应的初始排序序列包括N0个极化信道的索引,该N0个极化信道的索引是按照该N0个极化信道的可靠度的大小排序的,第一序列包括N个极化信道的索引,该N个极化信道的索引是按照该N个极化信道的可靠度的大小排序的,其中,N和N0为正整数,且N≤N0。
在一种可能的实现方式中,第一序列可以通过如下方式生成:获取该N个极化信道的极化权重,并将该N个极化信道的索引按照该N个极化信道的极化权重的大小进行排序,得到第一序列;或者,根据至少一个第一指示序列对第一基础序列进行扩展,得到第一序列,其中,每个第一指示序列用于将一个长度等于n的序列扩展为长度等于2n的序列,第一基础序列包括R个极化信道的索引,该R个极化信道的索引是按照该R个极化信道的可靠度大小排序的,其中,n、R均为正整数,R<N0。
在一种可能的实现方式中,第一基础序列可以通过如下方式生成:获取该R个极化信道的极化权重,并将该R个极化信道的索引按照该R个极化信道的极化权重的大小进行排序,得到第一基础序列。
在一种可能的实现方式中,该最大母码长度对应的初始排序序列可以通过如下方式生成:根据至少一个第二指示序列,对第二基础序列进行扩展得到该最大母码长度对应的初始排序序列,其中,第二基础序列包括T个极化信道的索引,该T个极化信道的索引是按照该T个极化信道的可靠度的大小排序的,每个第二指示序列用于将一个长度等于n的序列扩展为长度等于2n的序列,其中,n和T均正整数,T<N0。
第三方面,本申请提供一种极化码编码的装置,用于执行第一方面或第一方面的任意可能的实现方式中的方法。具体地,该装置包括执行第一方面或第一方面的任意可能的实现方式中的方法的单元。
第四方面,本申请提供一种译码的装置,用于执行第二方面或第二方面的任意可能的实现方式中的方法。具体地,该装置包括执行第二方面或第二方面的任意可能的实现方式中的方法的单元。
第五方面,本申请提供一种设备,该设备包括一个或多个处理器,一个或多个存储器,一个或多个收发器(每个收发器包括发射机和接收机)。发射机或接收机通过天线收发信号。存储器用于存储计算机程序指令(或者说,代码)。处理器用于执行存储器中存储的指令,当指令被执行时,处理器执行第一方面或第一方面的任意可能的实现方式中的方法。
第六方面,本申请提供一种设备,该设备包括一个或多个处理器,一个或多个存储器,一个或多个收发器(每个收发器包括发射机和接收机)。发射机或接收机通过天线收发信号。存储器用于存储计算机程序指令(或者说,代码)。处理器用于执行存储器中存储的指令,当指令被执行时,处理器执行第二方面或第二方面的任意可能的实现方式中的方法。
第七方面,本申请提供一种计算机可读存储介质,该计算机可读存储介质中存储有指令,当其在计算机上运行时,使得计算机执行上述第一方面或第一方面的任意可能的实现方式中的方法。
第八方面,本申请提供一种计算机可读存储介质,该计算机可读存储介质中存储有指令,当其在计算机上运行时,使得计算机执行上述第二方面或第二方面的任意可能的实现方式中的方法。
第九方面,本申请提供一种芯片(或者说,芯片系统),包括存储器和处理器,存储器用于存储计算机程序,处理器用于从存储器中调用并运行该计算机程序,使得安装有该芯片的通信设备执行上述第一方面和第二方面及其任一种可能的实现方式中的方法。
应理解这里的通信设备可以为第一方面的发送设备或第二方面的接收设备。
第十方面,本申请提供一种编码装置,该编码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发送设备行为的功能。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及其第一方面的任意一种可能的设计中的编码方法;输出接口电路,用于输出极化编码后的序列。
可选的,所述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,所述编码装置可以实现如上述第一方面及其第一方面的任意一种可能的设计中所述的编码方法。
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。
第十一方面,本申请提供一种译码装置,该译码装置具有实现上述第二方面和第二方面的任一种可能的设计中接收设备行为的功能。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功能上相对应的模块。
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,所述译码装置包括:输入接口电路,用于获取待译码序列;逻辑电路,用于执行上述第二方面及其第二方面任意一种可能的设计中的译码方法;输出接口电路,用于输出译码后的序列。
可选的,所述译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,所述译码装置包括: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,所述编码装置可以实现如上述第二方面及其第二方面任意一种可能的设计中所述的译码方法。
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,所述译码装置包括处理器。其中,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。
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。
本申请实施例提供的极化码编码方法和装置,通过将一些性能较优的排序序列进行融合,能够使得最终生成的最大母码长度对应的排序序列或者极化信道的可靠度的归一化序列的性能得到改善,从而能够提高极化码的译码性能。
附图说明
图1为本申请实施例所适用的无线通信系统。
图2是采用无线技术进行通信的基本流程图。
图3为极化码编码的处理流程的示意框图。
图4为本申请实施例提供的极化码编码的方法100的示意性交互图。
图5是基于基础序列和指示序列进行扩展的示意图。
图6为本申请实施例提供的极化码编码的装置300的示意性框图。
图7为本申请实施例提供的通信设备500的示意性结构图。
图8为处理装置504的一种内部结构示意图。
图9处理装置504的另一种内部结构示意图。
图10处理装置504的另一种内部结构示意图。
图11本申请实施例提供的极化码译码的装置600的示意性框图。
图12为本申请实施例提供的一种通信设备700的示意性结构图。
图13为处理装置704的一种内部结构示意图。
图14为处理装置704的另一种内部结构图。
图15为终端设备800的结构示意图。
具体实施方式
下面将结合附图,对本申请中的技术方案进行描述。
图1为本申请实施例所适用的无线通信系统。该无线通信系统中可以包括至少一个网络设备,该网络设备与一个或多个终端设备进行通信。该网络设备可以是基站,也可以是基站与基站控制器集成后的设备,还可以是具有类似通信功能的其它设备。
本申请实施例提及的无线通信系统包括但不限于:窄带物联网系统(NarrowBand-Internet of Things,NB-IoT)、全球移动通信系统(Global System for MobileCommunications,GSM)、增强型数据速率GSM演进系统(Enhanced Data rate for GSMEvolution,EDGE)、宽带码分多址系统(Wideband Code Division Multiple Access,WCDMA)、码分多址2000系统(Code Division Multiple Access,CDMA2000)、时分同步码分多址系统(Time Division-Synchronization Code Division Multiple Access,TD-SCDMA),长期演进系统(Long Term Evolution,LTE)、下一代5G移动通信系统的三大应用场景,即增强移动带宽(Enhance Mobile Broadband,eMBB),高可靠性低延迟通信(UltraReliable Low Latency Communication,URLLC)和增强海量机器连接通信(MassiveMachine Type Communication,eMTC)或者将来出现的新的通信系统。
本申请实施例中所涉及到的终端设备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它处理设备。终端可以是移动台(Mobile Station,MS)、用户单元(subscriber unit)、蜂窝电话(cellularphone)、智能电话(smart phone)、无线数据卡、个人数字助理(Personal DigitalAssistant,PDA)电脑、平板型电脑、无线调制解调器(modem)、手持设备(handset)、膝上型电脑(laptop computer)、机器类型通信(Machine Type Communication,MTC)终端等。
图1中的网络设备与终端之间采用无线技术进行通信。当网络设备发送信号时,其为发送端,当网络设备接收信号时,其为接收端。终端也是一样的,当终端发送信号时,其为发送端,当终端接收信号时,其为接收端。
图2是采用无线技术进行通信的基本流程图。发送端的信源依次经过信源编码、信道编码、速率匹配和调制后在信道上发出,接收端收到信号后依次经过解调、解速率匹配、信道解码和信源解码后获得信宿。
信道编解码是无线通信领域的核心技术之一,其性能的改进将直接提升网络覆盖及用户传输速率。目前,极化码是可理论证明达到香农极限,并且具有可实用的线性复杂度编译码能力的信道编码技术。极化码构造的核心是通过“信道极化”的处理,在编码侧,采用编码的方法使各个子信道呈现出不同的可靠性,当码长持续增加时,一部分信道将趋向于容量接近于1的无噪信道,另一部分信道趋向于容量接近于0的全噪信道,选择在容量接近于1的信道上直接传输信息以逼近信道容量。
Polar码的编码策略正是应用了这种现象的特性,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(也称为生成矩阵)为FN,编码过程为其中,是一个二进制的行矢量,长度为N(即,码长),且N=2n,n为正整数。FN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,以上各式中涉及的加法、乘法操作均为二进制伽罗华域上的加法、乘法操作。
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A。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(frozen bits)集合,其索引的集合用A的补集Ac表示。Polar码的编码过程相当于这里,FN(A)是FN中由集合A中的索引对应的行得到的子矩阵。FN(AC)是FN中由集合AC中的索引对应的行得到的子矩阵。uA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的uA为中的信息比特集合,uA为长度K的行矢量,即|A|=K,符号||表示集合中元素的个数,K为信息块大小,FN(A)是矩阵FN中由集合A中的索引对应的那些行得到的子矩阵,FN(A)是一个N×N的矩阵。
Polar码的构造过程即集合A的选取过程,它决定了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,这N个极化信道分别对应编码矩阵的N个行。计算极化信道的可靠度,将可靠度较高的前K个极化信道的序号(或者说,索引)作为集合A的元素,剩余(N-K)个极化信道对应的序号作为固定比特序号集合AC的元素。集合A决定了信息比特的位置,集合AC决定了固定比特的位置。
下面对本申请实施例提供的极化码编码的方法进行详细说明。
首先,对极化码编码的基本处理流程进行说明。
参见图3,图3为极化码编码的处理流程的示意框图。如图3所示,Polar码在编码前首先利用Polar码的构造模块选出信息比特的索引集合I。其中,Polar码构造模块可以基于在线计算,也可以基于读表的方式获得每个极化信道的可靠度排序,然后根据信息比特的长度K选择可靠度最高的K个极化信道作为信息比特位置,其余作为固定比特。一般地,采用在线读表的方式,即可以根据采用离线计算得到最大母码长度对应的排序序列并存表。在进行极化编码时,根据当前母码长度从最大母码长度对应的排序序列中读取当前所需母码长度对应的排序序列。最后根据信息比特长度从排序序列中确定信息比特的位置集合完成Polar码的构造过程。
下面对本申请实施例提供的极化码编码的方法进行说明。
参见图4,图4为本申请实施例提供的极化码编码的方法100的示意性交互图。
110、发送设备获取所需母码长度对应的排序序列。
这里所说的所需长度是针对信息比特长度K而言的。在进行极化编码时,不同的信息比特长度K,编码时需要选取不同长度的排序序列。
例如,若K=7,则所需母码长度=8。若K=9,则所需母码长度=16。母码长度的取值均为2的整数次幂。
120、发送设备获取待编码比特。
130、发送设备采用该所需母码长度对应的排序序列对待编码比特进行极化编码,得到编码后的序列。
发送设备获取到所需母码长度对应的排序序列后,根据该所需母码长度对应的排序序列确定出信息比特索引集合。再根据信息比特索引集合对待编码比特进行极化编码,得到编码后的序列。
选取信息比特索引集合,即是从所需母码长度对应的排序序列包括的N个极化信道的索引中,选择可靠度最大的前K个极化信道的索引,作为极化编码时放置信息比特的位置。其中,选取信息比特索引集合的过程可以与现有技术相同。
假定所需的母码长度等于N,即是说该所需母码长度对应的排序序列包括N个极化信道的索引(也称作序号),该N个索引与N个极化信道一一对应。该排序序列中的N个索引是按照该N个极化信道的可靠度的大小排序的。不失一般性地,该排序序列可以是按照该N个极化信道的可靠度从小到大的顺序排列的,或者,也可以是按照这N个极化信道的可靠度从大到小排列的。
极化信道的可靠度可以有多种参数衡量。例如,错误概率、信道容量或极化权重等,或者,还可以是能够衡量极化信道的可靠度的其它参数。
在本申请实施例中,假定极化编码时所需的母码长度对应的排序序列的长度等于N,则该排序序列可以是从一个长度为N0的序列中直接读取的,N≤N0,且N和N0均为正整数。
需要说明的是,这里的长度为N0的序列可以是系统所能支持或事先约定的最大母码长度对应的排序序列,也可以是任意长度的母码。系统能支持或事先约定的母码长度可以为512或者1024等。。
其中,该长度为N0的序列包括N0个极化信道的索引,这N0个极化信道的索引是按照这N0个极化信道的可靠度的大小顺序排列的。
因此,发送设备可以先获取这个长度为N0的序列,并将该序列进行存储。在进行极化编码时,根据信息比特长度K,每次都直接从该长度为N0的序列中直接读取所需母码长度对应的排序序列即可。
例如,假设长度为N0=16的序列为{0,1,2,4,8,9,3,5,10,12,6,11,7,13,14,15},由此可以从该长度为16的序列中读取长度为8的序列,按照索引号为0-7的索引在该长度为16的序列中的位置顺序读取得到长度为8的序列为{0,1,2,4,3,5,6,7}。
应理解,如果所需的母码长度等于N,则从该长度为N0的序列中依次读取小于或等于N的索引,这些索引构成的序列即为该所需母码长度对应的排序序列。
需要说明的是,如果长度为N0的序列中的索引是从0开始,则依次读取小于N的索引。如果长度为N0的序列中的索引是从1开始,则依次读取小于或等于N的索引。
在本申请实施例中,长度为N0的序列可以通过多种方式生成。下面,对如何生成该长度为N0的序列的过程进行说明。
为了便于理解本申请实施例提供的技术方案,首先对几种现有的构造polar码的最大母码对应的排序序列的方案作简单介绍。
方案1
采用搜索算法,合成满足嵌套特性的最大母码长度对应的排序序列。
例如,根据最大母码长度,计算不同信息比特个数K对应的可靠度最高的信息比特索引集合。针对每个K,可以根据当前的码长、码率,采用搜索算法或者不同的构造算法得到可靠度最高的信息比特索引集合。再根据不同K对应的信息比特索引集合的差异,逐比特合成最大母码长度对应的排序序列。
需要说明的是,嵌套特性是指从一个长度为较大的排序序列中读取一个长度较小的排序序列,该长度较小的排序序列是该长度较大的排序序列的子集。满足这种特性的两个排序序列即满足嵌套特性。
方案2
根据极化权重的计算公式,构造最大母码长度对应的排序序列。
极化权重的计算公式也称作PW公式,具体可以如下公式(1)所示:
其中,Wi为第i个极化信道的极化权重,Bj∈{0,1},j∈{0,1,...,n-1},i=Bn- 1Bn-2...B0,Bn-1Bn-2...B0为i的二进制表示。权重修正参数φ和α一般分别取值0和1/4。
根据公式(1)计算最大母码长度N对应的N个极化信道的极化权重,并按照计算得到的极化权重的大小对该N个极化信道的索引进行排序,得到最大母码长度对应的排序序列。
在实际使用时,也可以使用公式(1)的变形形式,例如如下的公式(2)所示:
其中,(a,b,c)一般取值为(1,0,1/4)或(1.07,0.5,1/4)。
通过方案2构造得到最大母码长度对应的排序序列也满足嵌套关系。
通常,编码端都只存储该最大母码长度对应的排序序列,在进行编码时,编码端只需要从该最大母码长度对应的排序序列中依次读取小于当前所需母码长度的索引,即为当前所需母码长度对应的排序序列。
方案3
给定一个基础序列,根据至少一个指示序列对基础序列进行逐级扩展,直至扩展到最大母码长度对应的排序序列。
参见图5,图5是基于基础序列和指示序列进行扩展的示意图。如图5所示,基础序列是码长等于8的排序序列{0,1,2,4,3,5,6,7}。根据排序序列之间的嵌套特性可以得到码长等于16时,索引小于8的极化信道的可靠度的排序序列为{0,1,2,4,3,5,6,7}(以下记作排序序列#1)。再根据排序序列的对称性,可以得到索引大于8的极化信道的可靠度的排序序列为{0+8,1+8,2+8,4+8,3+8,5+8,6+8,7+8}(以下记作排序序列#2)。另外,从8扩展到16的指示序列为{1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0}。
我们将这两组排序序列中的索引相互之间的排序记作排序序列#3。
其中,指示序列中的元素0表示将排序序列#2中的元素添加到排序序列#3,指示序列中的元素1表示将排序序列#1中的元素添加到排序序列#3。将初始的排序序列#3视作空集,通过指示序列中的元素依次从排序序列#1或排序序列#2中读取极化信道的索引加入到排序序列#3,得到最终的排序序列#3。最终的排序序列#3为{0,1,2,4,0+8,1+8,3,5,2+8,4+8,6,3+8,7,5+8,6+8,7+8}={0,1,2,4,8,9,3,5,10,12,6,11,7,13,14,15}。
采用方式3时,指示序列可以是一组离线生成的序列,来指示不同级别的扩展,其生成方式,可以通过不同的算法得到。
该方案的缺点是需要存多个指示序列,且从较短的基础序列经过多级扩展会引起性能的下降。
方案4
根据距离谱计算极化信道的可靠度排序,生成最大母码长度对应的排序序列。
为了便于理解,这里对距离谱的概念作简单介绍。
首先给出0陪集和1陪集的定义。
给定已译码序列表示序列{u0,u1,…,ui-1},并且,即中的元素为0或1。
0陪集定义为:
1陪集定义为:
陪集间的距离谱
对于N=2m,0陪集和1陪集间的距离谱为其定义为:
陪集内部距离谱
对于N=2m,0陪集内部距离谱为
由于译码准确性与当前译码比特对应的码字集合(陪集Coset)的间距和距离谱相关。其中,间距或者距离谱越近,该极化信道的可靠度越高。
基于距离谱的定义,对于第i个极化信道的可靠度的定义可以表示为公式(7):
其中,在上述公式(3)至公式(7)中出现的参数的含义如下:
表示0陪集,span(·)表示向量生成的线性空间,gi表示Polar码的生成矩阵中第i行向量。
表示1陪集,表示陪集间距离谱,表示陪集内部距离谱。表示0向量,表示标号为0~(i-1)的译码比特均为0。W表示码字的重量,或者生成矩阵的行重。重量即码字中包含的1的个数,w∈{0,1...,N}。p∈(0,1/2)。L(m)(i,p)表示第i个极化信道的可靠度。
关于陪集的更详细的概念可以参见现有技术,这里不再作详细介绍。
因此,在方案4中,可以基于公式(7)计算N个极化信道的可靠度,并将该N个极化信道的索引按照可靠度的大小进行排序,即可得到排序序列。当N取最大母码长度时,可以得到最大母码长度对应的排序序列。
该方案的缺点是计算复杂度很高,尤其在需要确定最大母码长度对应的排序序列时,随着码长的增大,复杂度也随之增加。
鉴于以上各方案各自存在的缺陷,本申请实施例还提供了更优的确定最大母码长度对应的排序序列的方式,可以融合以上各方案的优点,可以提高极化码的译码性能。同时也具有降低计算复杂度、减小存储开销等优点。
下面对本申请实施例提供的几种生成长度为N0的序列的方式进行说明。
方式1
可选地,长度为N0的序列是通过将最大母码长度对应的初始排序序列中小于N的索引依次替换为第一序列包括的索引得到的。
其中,最大母码长度对应的初始排序序列包括N0个极化信道的索引,该N0个极化信道的索引是按照该N0个极化信道的可靠度的大小排序的,第一序列包括N个极化信道的索引,该N个极化信道的索引是按照该N个极化信道的可靠度的大小排序的,其中,N和N0为正整数,且N≤N0。
需要注意的是,这里所说的最大母码长度对应的初始排序序列与长度为N0的排序序列相互区别。
其中,最大母码长度对应的初始排序序列长度等于N0,可以通过现有技术中计算极化信道可靠度的排序序列的方法生成,例如,可以采用前文所述的方案1至方案4中的任意一种方法生成。在该初始排序序列的基础上,通过一个长度较小的排序序列(记作短排序序列)替换该初始排序序列中与该短排序序列相等码长的子序列,替换后的序列即为码长等于N0的排序序列。
举例来说,假定N0=256,根据前文所述,我们已经知道排序序列之前满足嵌套特性,因此,码长等于8、16、32、64、128的排序序列都可以从该码长等于256的排序序列中直接读取得到。
采用码长等于64(或者8、16、32、128)的短排序序列将码长等于256的排序序列中码长等于64的序列替换掉,得到一个新的码长等于256的排序序列。
在这个示例中,替换之前的码长等于256的排序序列对应初始排序序列,码长等于64的短排序序列对应第一序列。
具体实现时,可以将初始排序序列中小于或等于N的索引依次替换为第一序列中的索引。例如,最大母码长度对应的初始排序序列为{0,1,2,4,3,5,6,7},N0=8。若第一序列为{0,1,2,3},N=4。则初始排序序列中小于4的索引对应的位置分别是第1个位置、第2个位置、第3个位置和第5个位置。将这些位置的索引依次替换为第一序列中的索引,即,一一对应替换为0,1,2,3,得到的序列则为该长度为N0(N0=8)的序列。
需要说明的是,在这个示例中,最大母码长度对应的初始排序序列中小于N的索引的相对顺序与第一序列中索引的相对顺序是相同的。因此,替换之后的序列与替换之前的序列相同。在另一些情况下,初始排序序列中小于N的索引的相互顺序与第一序列中索引的相互顺序不同,替换之后的序列与替换之前的序列则不同。
需要说明的是,为了使得替换后的排序序列的性能更好。不失一般性地,生成最大码长对应的初始排序序列的方法与生成第一序列的方法应不同。
与初始排序序列类似,第一序列也可以采用上述方案1至4中的任意一种方法生成。
例如,采用前文所述的方案2生成最大母码长度对应的初始排序序列,采用方案4生成第一序列。或者,还可以是其它组合。
下面给出采用上述方案4生成的第一序列。其中,具体给出码长等于256和码长等于512两种情况。
第一序列的长度N=256
{0,1,2,4,8,16,32,3,64,5,128,6,9,10,17,12,18,33,20,34,24,36,65,7,40,66,68,11,48,129,72,13,14,19,130,80,132,21,22,35,25,136,96,144,26,37,28,38,160,192,67,41,42,69,49,15,44,70,50,131,73,23,52,74,133,81,56,76,134,82,27,39,137,29,97,84,138,98,43,145,30,140,88,146,100,71,45,51,46,148,161,104,162,152,112,75,53,164,193,54,77,57,83,78,135,194,31,58,168,85,139,196,176,60,86,99,141,89,200,147,47,101,142,90,208,102,149,92,105,224,150,106,163,55,153,113,79,165,108,154,59,114,166,156,195,87,61,116,169,62,143,170,91,197,177,198,120,172,178,201,103,93,202,180,151,209,94,107,204,184,155,109,115,210,167,110,225,157,212,63,158,117,226,171,118,216,121,199,173,228,179,174,122,95,232,203,124,181,240,205,182,185,211,111,206,186,159,213,188,119,214,227,217,175,229,218,123,230,220,125,233,183,234,126,207,187,241,236,215,242,189,244,190,219,248,231,221,127,235,222,237,243,238,245,191,246,249,223,250,252,239,247,251,253,254,255}。
第一序列的长度N=512
{0,1,2,4,8,16,32,64,3,5,6,128,9,256,10,17,12,18,20,33,34,24,65,36,7,40,66,68,11,48,13,129,72,130,14,19,21,80,132,136,22,96,25,35,26,257,144,37,258,28,38,260,41,42,67,44,69,49,70,15,73,50,131,23,74,264,160,52,81,133,76,27,39,272,134,82,137,56,29,192,259,43,30,97,288,138,84,145,98,261,71,140,45,88,51,320,262,46,146,100,265,75,161,53,384,148,266,104,162,77,54,273,135,152,268,57,83,112,193,78,164,274,31,58,289,194,139,85,276,168,60,290,196,99,86,176,141,280,89,321,263,47,147,101,292,200,142,90,322,296,102,208,92,149,105,267,385,324,55,163,150,106,304,224,79,153,269,113,59,386,165,328,275,108,154,270,114,166,87,195,61,156,169,277,388,116,336,62,91,143,291,197,278,170,281,120,392,177,352,198,172,103,93,293,201,282,178,151,400,294,323,202,284,94,416,180,297,448,209,204,184,298,325,210,107,305,155,109,300,326,225,271,212,115,387,306,329,167,110,226,216,157,330,63,308,117,158,389,171,279,118,121,199,337,228,173,179,332,283,390,95,312,122,338,174,232,393,295,203,285,124,181,299,205,353,286,340,182,394,401,211,354,185,206,301,396,344,186,240,327,111,159,402,356,417,213,307,404,360,188,302,119,418,214,227,331,217,309,175,229,310,123,333,408,218,391,313,368,449,339,334,420,230,450,125,220,233,314,424,183,287,395,341,126,234,452,432,316,207,355,241,342,187,236,397,345,403,303,357,242,346,398,215,189,358,456,244,464,348,405,190,361,311,419,406,219,248,335,362,409,480,231,369,221,315,421,410,364,127,222,370,422,412,451,235,425,317,343,318,372,426,237,399,347,453,243,376,238,433,454,359,428,191,457,434,349,245,458,436,350,407,246,363,411,465,249,460,365,223,423,440,371,250,366,466,413,252,481,468,319,373,414,482,472,427,374,239,429,377,455,435,430,378,459,351,484,437,247,380,488,496,438,461,441,251,467,462,367,415,442,253,469,483,375,254,470,444,473,431,485,379,474,486,476,489,381,439,463,490,382,497,492,443,255,445,498,471,446,500,504,475,487,477,383,491,478,493,499,494,501,502,505,506,508,447,479,495,503,507,509,510,511}。
在方式1中,生成最大母码长度对应的排序序列时,可以采用前文的方案2,计算极化信道极化权重,并按照极化权重的大小生成初始排序序列。
在采用方案2时,公式(1)中涉及到一些参数(例如,a,b,c)的取值。当这些参数取值不同时,生成的排序序列也会不同。
参见表1,表1是公式(1)中涉及的参数的取值与第一序列的长度的取值的几种组合。在每一种组合下,生成的长度为N0的序列也互不相同。下面分别给出。
表1
组合 | 参数 |
1 | a=1,b=0,c=1/4,N=256,N<sub>0</sub>=1024 |
2 | a=1,b=0,c=1/4,N=512,N<sub>0</sub>=1024 |
3 | a=1.07,b=0.5,c=1/4,N=256,N<sub>0</sub>=1024 |
4 | a=1.07,b=0.5,c=1/4,N=512,N<sub>0</sub>=1024 |
以下,分别将表1中4种组合下生成的长度为N0的序列分别记作序列1、序列2、序列3和序列4。
序列1
{0,1,2,4,8,16,32,3,5,64,6,9,10,128,17,12,18,33,20,34,24,65,36,66,7,256,40,129,11,68,48,13,130,19,14,72,21,132,22,80,512,35,25,136,96,37,257,144,26,38,67,258,28,41,160,42,44,49,260,69,192,15,70,131,73,264,50,23,513,74,52,133,81,76,514,134,27,272,56,82,137,39,516,259,97,29,84,138,145,30,98,288,43,261,88,520,140,146,100,71,45,262,161,51,265,148,46,528,104,162,320,266,75,53,152,515,112,193,273,164,77,54,268,135,544,57,517,274,83,194,78,168,58,85,518,384,289,60,86,276,139,99,521,196,89,141,176,290,576,263,90,101,522,147,142,280,31,102,529,47,292,200,321,524,267,149,92,105,208,530,163,150,322,55,296,106,269,153,640,545,224,532,275,113,79,165,324,108,270,154,546,59,519,385,304,114,166,277,87,195,536,61,156,328,548,386,291,169,577,278,197,170,116,523,177,91,281,62,143,768,578,388,293,198,103,552,336,172,525,282,201,120,178,93,531,151,294,580,323,526,392,297,202,107,284,180,641,560,209,533,94,352,204,325,298,584,271,184,210,642,547,109,534,400,305,155,115,110,326,537,167,300,212,329,644,549,387,306,157,592,279,225,117,171,538,226,216,330,550,416,769,158,579,389,308,118,648,173,553,337,121,540,283,199,179,332,228,770,608,390,295,174,581,554,338,122,527,393,312,232,203,285,181,656,561,124,772,582,448,353,556,394,340,299,182,585,286,205,63,643,562,185,535,401,211,354,206,327,776,586,396,301,95,186,672,344,645,564,402,307,240,593,213,111,356,539,159,302,588,331,646,551,417,784,227,594,404,309,214,649,119,568,188,541,217,360,333,418,771,704,609,391,310,229,650,596,555,339,218,542,408,313,175,123,334,657,800,610,420,773,230,583,449,368,652,557,395,341,314,220,600,287,233,125,658,563,183,774,612,450,355,558,424,342,777,234,587,397,316,241,207,673,345,660,565,403,187,832,452,236,357,778,616,398,303,126,589,674,346,647,566,432,785,242,595,405,215,664,358,569,189,780,590,456,361,676,348,419,786,705,624,406,311,244,651,597,570,219,543,409,190,362,335,896,801,706,611,421,788,231,680,598,464,369,653,572,410,315,248,601,221,364,659,802,422,775,708,613,451,370,654,559,425,343,792,235,602,412,317,222,237,688,661,804,614,480,833,453,426,372,779,712,617,399,318,243,604,675,347,662,567,433,238,834,454,665,359,808,618,428,781,245,591,457,376,677,349,434,787,720,625,407,127,666,836,571,246,782,620,458,363,678,350,897,816,707,626,436,789,249,681,599,465,668,573,411,191,840,460,365,898,803,736,423,790,709,682,628,466,371,655,574,440,793,250,603,413,252,366,689,900,805,710,615,481,848,684,468,427,373,794,713,632,414,319,223,605,690,663,806,482,835,455,904,374,809,714,619,429,796,239,606,472,377,692,435,721,864,484,667,837,810,430,783,716,621,459,378,679,351,912,817,722,627,437,247,696,838,669,812,622,488,841,461,380,899,818,737,438,791,724,683,629,467,670,575,441,251,842,462,367,928,738,901,820,711,630,496,849,685,469,442,795,728,633,415,253,844,691,902,807,740,483,850,686,470,905,375,824,715,634,444,797,254,607,473,693,960,865,485,906,852,811,744,431,798,717,636,474,379,694,913,723,866,486,697,839,908,813,718,623,489,856,476,381,914,819,752,439,725,698,868,671,814,490,843,463,382,929,739,916,821,726,631,497,700,443,729,872,492,845,930,903,822,741,498,851,687,471,920,825,730,635,445,255,846,932,742,961,880,500,907,853,826,745,446,799,732,637,475,695,962,867,487,936,854,746,909,828,719,638,504,857,477,915,753,964,699,869,910,815,748,491,858,478,383,944,754,917,727,870,701,968,873,493,860,931,918,823,756,499,702,921,731,874,494,847,933,743,976,881,501,922,827,760,447,733,876,934,963,882,502,937,855,747,924,829,734,639,505,992,965,938,884,911,830,749,506,859,479,945,755,966,871,940,750,969,888,508,861,946,919,757,703,970,875,495,862,948,758,977,923,761,972,877,935,978,883,503,952,762,925,735,878,993,980,939,885,926,831,764,507,994,967,886,941,751,984,889,509,947,996,942,971,890,510,863,949,759,1000,973,892,950,979,953,763,974,879,1008,981,954,927,765,995,982,887,956,766,985,997,943,986,891,511,998,1001,988,893,951,1002,975,894,1009,955,1004,1010,983,957,767,1012,958,987,999,1016,989,1003,990,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015”,”1019,1021,1022,1023}
序列2
{0,1,2,4,8,16,32,3,5,64,6,9,10,128,17,12,18,33,256,20,34,24,65,36,66,7,40,129,11,68,48,13,130,19,14,72,257,21,132,22,512,80,35,258,25,136,96,260,37,144,26,38,264,67,28,41,160,42,44,49,69,272,192,15,70,131,73,50,513,288,23,74,52,133,514,81,320,76,134,27,56,82,516,137,259,39,97,384,29,84,138,261,145,30,98,520,43,88,140,262,146,100,265,71,45,161,51,528,148,46,104,266,162,273,75,515,53,152,112,268,193,164,274,77,544,54,517,135,57,83,194,289,78,168,518,276,58,85,60,86,139,99,521,196,89,290,280,141,576,176,90,101,522,147,142,31,102,263,529,47,321,292,200,524,322,149,296,92,105,530,208,267,385,324,304,163,150,55,640,545,106,532,153,224,386,328,269,113,79,165,546,108,519,154,275,59,270,114,166,87,536,195,61,156,548,169,277,291,577,278,197,170,116,523,177,281,91,62,143,768,578,388,293,198,103,552,336,172,525,282,201,120,178,294,531,93,151,580,323,526,392,297,202,107,284,180,641,560,209,533,94,204,298,352,325,584,184,210,400,642,547,305,534,300,109,155,326,115,110,537,167,212,306,329,644,549,157,225,117,592,171,330,226,387,538,308,216,416,550,337,769,158,579,271,118,279,648,332,553,389,173,540,121,199,179,228,338,770,608,312,390,174,581,554,393,283,527,122,232,340,448,353,394,656,561,203,772,582,181,295,556,285,124,182,205,585,63,286,299,643,562,354,535,185,401,211,396,344,776,586,206,301,95,186,672,240,645,564,327,402,213,593,356,307,302,539,111,159,588,417,646,551,331,784,227,594,404,309,214,649,119,568,188,541,360,418,408,368,771,704,609,217,449,420,650,596,555,310,229,542,333,218,175,391,123,657,800,610,313,773,230,583,339,334,652,557,220,450,424,314,600,233,125,287,658,563,183,774,612,341,395,558,355,342,777,234,587,397,316,345,241,673,207,660,565,403,357,832,187,236,126,778,616,303,452,432,589,674,242,647,566,346,785,398,595,215,405,664,358,569,361,780,590,189,456,676,348,419,786,705,624,406,244,409,651,597,570,362,543,219,421,369,311,896,801,706,611,190,788,410,680,598,231,248,653,572,364,464,335,601,422,315,659,802,221,775,708,613,370,425,654,559,451,412,792,235,602,222,343,372,426,688,661,804,614,480,833,453,317,237,779,712,617,433,347,318,604,675,454,662,567,243,428,834,399,665,359,808,618,238,781,376,591,457,434,677,349,245,787,720,625,458,407,666,836,571,127,782,620,363,350,678,246,897,816,707,626,436,789,465,681,599,411,668,573,460,249,840,365,466,898,803,736,423,790,709,682,628,191,371,655,574,440,793,250,603,413,366,468,689,900,805,710,615,481,848,684,373,427,414,794,713,632,252,223,374,605,690,663,806,482,835,429,904,455,809,714,619,472,796,377,606,435,319,692,239,721,864,430,667,837,810,484,783,716,621,459,378,679,437,912,817,722,627,488,461,696,838,669,812,622,380,841,438,351,899,818,737,247,791,724,683,629,467,670,575,441,251,842,462,496,928,738,901,820,711,630,442,849,685,367,469,795,728,633,470,415,844,691,902,807,740,483,850,686,253,905,444,824,715,634,375,797,473,607,485,693,960,865,474,906,852,811,744,431,798,717,636,379,486,694,913,723,866,254,697,839,908,813,718,623,476,856,489,439,914,819,752,490,725,698,868,671,814,381,843,463,382,929,739,916,821,726,631,497,700,443,729,872,492,845,930,903,822,741,498,851,687,445,920,825,730,635,471,446,846,932,742,961,880,475,907,853,826,745,500,799,732,637,487,695,962,867,504,936,854,746,909,828,719,638,255,857,477,915,753,964,699,869,910,815,748,491,858,478,493,944,754,917,727,870,701,968,873,499,860,931,918,823,756,494,702,921,731,874,501,847,933,743,976,881,383,922,827,760,447,733,876,934,963,882,502,937,855,747,924,829,734,639,505,992,965,938,884,911,830,749,506,859,479,945,755,966,871,940,750,969,888,508,861,946,919,757,703,970,875,495,862,948,758,977,923,761,972,877,935,978,883,503,952,762,925,735,878,993,980,939,885,926,831,764,507,994,967,886,941,751,984,889,509,947,996,942,971,890,510,863,949,759,1000,973,892,950,979,953,763,974,879,1008,981,954,927,765,995,982,887,956,766,985,997,943,986,891,511,998,1001,988,893,951,1002,975,894,1009,955,1004,1010,983,957,767,1012,958,987,999,1016,989,1003,990,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列3
{0,1,2,4,8,16,32,3,5,64,6,9,10,128,17,12,18,33,20,34,24,65,36,66,256,7,40,129,11,68,48,13,130,19,14,72,21,132,22,512,80,35,25,136,96,37,144,257,26,38,67,28,258,41,160,42,260,44,49,69,192,15,70,264,131,73,50,23,74,513,52,133,81,514,272,76,134,27,56,82,137,39,516,97,29,84,138,259,288,145,30,520,98,43,88,140,261,146,100,71,45,262,161,528,51,265,148,320,46,104,162,75,266,53,152,112,193,515,273,544,268,164,77,54,135,57,83,274,194,517,78,384,168,58,85,60,518,276,289,86,139,99,521,196,576,89,290,141,176,522,280,90,101,147,142,31,263,292,529,102,47,524,200,149,321,92,530,105,267,208,322,296,163,640,150,55,106,153,532,545,269,324,224,113,79,165,275,108,546,304,270,154,59,385,536,114,166,328,87,195,519,277,61,548,386,156,169,197,577,170,278,291,116,768,177,523,388,281,578,336,552,91,62,143,198,103,172,293,282,201,525,120,178,580,392,294,93,531,151,560,526,284,202,323,297,107,352,641,180,209,584,94,533,298,204,642,400,325,184,210,109,534,155,547,305,271,326,300,115,644,537,110,592,167,212,329,306,157,225,549,387,538,416,117,171,330,226,279,648,216,550,308,769,158,118,389,540,173,579,337,553,608,332,121,770,199,179,228,390,283,338,554,312,174,656,581,122,393,232,295,448,772,203,561,527,285,181,582,340,556,124,353,394,182,205,585,562,286,299,63,776,185,354,643,401,672,396,586,344,211,206,535,95,186,564,402,327,301,356,645,593,240,213,588,307,111,784,302,646,539,404,417,594,568,159,331,227,214,360,649,119,551,309,704,418,188,541,217,596,609,650,408,333,310,771,229,800,218,391,542,339,555,420,313,610,368,657,334,175,652,123,230,449,600,773,314,220,658,233,583,341,557,395,612,450,424,774,125,563,287,183,342,558,316,777,234,355,660,832,673,397,241,587,452,345,616,207,187,565,778,403,236,674,432,357,398,346,126,664,589,242,566,785,303,456,780,358,647,405,595,676,569,624,590,348,215,786,189,361,244,705,406,419,570,896,597,219,362,651,409,311,706,464,680,788,801,190,543,231,421,598,572,611,369,410,335,364,653,248,802,708,601,422,315,221,792,370,659,688,613,654,412,451,425,602,775,480,804,235,222,343,559,317,712,614,372,661,833,426,237,453,604,617,318,779,662,834,243,808,675,433,399,454,347,428,618,376,665,238,567,720,245,457,781,434,359,836,677,625,666,591,349,787,620,127,458,816,782,407,678,571,436,897,626,350,246,363,668,840,249,707,465,681,789,736,460,898,191,599,573,411,628,466,682,440,790,365,803,709,250,423,574,900,793,371,848,689,366,655,413,710,603,468,684,481,632,805,252,794,223,713,690,615,373,414,427,904,482,806,605,319,714,472,796,374,663,835,809,692,864,455,429,606,619,484,377,721,239,810,716,435,912,837,430,378,667,722,696,621,459,817,783,488,838,812,679,437,627,351,622,380,669,841,247,818,724,737,461,438,899,928,629,670,842,467,683,441,791,738,496,462,820,251,575,728,901,630,849,442,367,844,711,469,685,740,633,902,795,850,253,824,691,415,470,686,444,905,483,634,807,960,254,715,473,797,375,744,852,693,865,906,607,485,636,474,798,811,717,694,866,913,431,908,486,379,856,723,697,752,718,476,914,489,839,813,868,698,623,381,819,725,490,814,439,916,929,382,671,843,726,700,872,739,497,463,821,492,930,729,631,443,920,845,498,822,741,730,903,851,932,825,880,846,471,687,445,742,635,500,961,255,826,732,745,853,446,907,962,936,637,475,799,746,504,854,828,695,867,909,487,638,964,857,753,719,477,915,748,944,869,910,858,699,754,478,491,815,968,870,917,383,860,727,701,873,756,918,493,931,702,874,921,499,823,976,494,731,760,933,881,922,847,876,743,501,934,827,733,882,447,924,502,963,937,992,734,747,505,855,829,884,938,639,965,506,830,749,945,911,966,859,940,888,755,479,750,508,969,946,871,861,757,970,919,948,862,703,875,758,977,495,972,761,923,978,952,877,762,935,883,878,925,503,980,993,735,764,885,926,939,994,507,831,984,886,967,941,996,889,751,509,947,942,890,510,971,1000,949,863,892,759,973,950,979,953,1008,974,763,954,879,981,765,927,982,956,995,766,985,887,997,986,943,998,891,511,988,1001,893,1002,951,894,1009,975,1004,955,1010,983,957,1012,767,958,987,1016,999,989,990,1003,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列4
{0,1,2,4,8,16,32,3,5,64,6,9,10,128,17,12,18,33,256,20,34,24,65,36,66,7,40,129,11,68,48,13,130,19,14,72,257,21,132,512,22,80,35,258,25,136,96,260,37,144,26,38,264,67,28,41,160,42,44,49,69,272,192,15,70,131,73,50,288,513,23,74,52,514,133,81,320,76,134,27,56,82,516,137,259,39,97,384,29,84,138,520,261,145,30,98,43,88,140,262,146,100,265,528,71,45,161,51,148,46,104,266,162,273,75,53,152,515,112,544,268,193,164,274,77,54,135,57,83,517,194,289,78,168,276,58,518,85,60,86,139,99,521,196,576,89,290,280,141,522,176,90,101,147,142,31,102,263,529,47,321,524,292,200,322,149,530,296,92,105,208,267,385,640,324,304,163,150,532,545,55,106,153,224,386,328,269,113,546,79,165,108,154,275,536,59,270,114,166,87,519,195,61,548,156,169,277,291,577,278,197,170,116,768,177,523,281,91,578,62,552,143,388,293,198,103,336,172,282,201,525,120,178,580,294,93,151,531,323,560,526,392,297,202,107,284,180,641,209,94,584,204,533,298,352,642,325,184,210,400,305,534,300,547,109,155,326,115,110,644,537,167,592,212,306,329,157,225,117,549,171,538,330,226,387,308,216,416,648,337,550,158,769,271,118,279,540,332,579,389,553,608,173,121,770,199,179,228,338,312,390,554,174,393,656,581,283,122,232,340,448,772,353,561,527,394,203,582,181,556,295,285,124,182,205,585,562,63,286,299,776,354,185,643,401,672,211,586,396,344,206,535,301,95,564,186,240,327,402,645,593,213,356,588,307,302,784,111,646,539,159,417,594,568,331,227,404,309,214,649,119,551,188,704,360,418,541,408,596,609,650,368,217,449,771,420,800,310,229,542,333,555,218,175,610,391,657,123,313,652,230,339,334,600,773,220,450,658,424,583,314,557,233,612,125,287,774,183,563,341,395,355,558,342,777,234,397,660,832,673,316,345,587,241,207,616,403,357,565,778,187,236,674,126,303,452,432,242,664,589,346,566,785,398,215,780,405,647,358,595,676,569,624,590,361,189,786,456,348,419,705,406,244,570,896,597,409,362,651,219,421,706,369,680,788,801,311,543,190,410,598,572,611,231,248,364,464,653,335,802,708,601,422,315,221,792,370,659,688,613,654,425,451,412,602,775,235,804,222,343,372,559,426,712,614,480,661,833,453,317,237,604,617,433,779,662,834,347,808,675,318,454,243,428,399,618,359,665,238,567,720,376,457,781,434,349,836,677,625,666,591,245,787,620,458,407,816,782,127,678,571,363,897,626,350,246,436,668,840,465,707,411,681,789,736,460,898,249,599,573,365,628,466,682,423,790,191,803,709,371,440,574,900,793,250,848,689,413,655,366,710,603,468,684,481,632,805,373,794,427,713,690,615,414,252,223,904,374,806,605,482,714,429,796,455,663,835,809,692,864,472,377,606,619,435,319,721,239,810,716,430,912,837,484,459,667,722,696,621,378,817,783,437,838,812,679,488,627,461,622,380,669,841,438,818,724,737,351,247,899,928,629,670,842,467,683,441,791,738,251,462,820,496,575,728,901,630,849,442,367,844,711,469,685,740,633,902,795,850,470,824,691,415,483,686,253,905,444,634,807,960,375,715,473,797,485,744,852,693,865,906,607,474,636,431,798,811,717,694,866,913,379,908,486,254,856,723,697,752,718,476,914,489,839,813,868,698,623,439,819,725,490,814,381,916,929,463,671,843,726,700,872,739,382,497,821,443,930,729,631,492,920,845,498,822,741,730,903,851,932,825,880,846,445,687,471,742,635,446,961,475,826,732,745,853,500,907,962,936,637,487,799,746,504,854,828,695,867,909,255,638,964,857,753,719,477,915,748,944,869,910,858,699,754,491,478,815,968,870,917,493,860,727,701,873,756,918,499,931,702,874,921,494,823,976,501,731,760,933,881,922,847,876,743,383,934,827,733,882,447,924,502,963,937,992,734,747,505,855,829,884,938,639,965,506,830,749,945,911,966,859,940,888,755,479,750,508,969,946,871,861,757,970,919,948,862,703,875,758,977,495,972,761,923,978,952,877,762,935,883,878,925,503,980,993,735,764,885,926,939,994,507,831,984,886,967,941,996,889,751,509,947,942,890,510,971,1000,949,863,892,759,973,950,979,953,1008,974,763,954,879,981,765,927,982,956,995,766,985,887,997,986,943,998,891,511,988,1001,893,1002,951,894,1009,975,1004,955,1010,983,957,1012,767,958,987,1016,999,989,990,1003,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
这里给出的序列1至序列4,在不同的仿真参数下,与包括前文所述的方案1至方案4在内的现有生成极化信道可靠度的排序序列的方法相比较,具有更好的性能。
在新一代无线接入(New Radio Access,NR)系统的标准中规定,NR采用的序列需要满足嵌套特性。并且,一个序列的性能的好坏可以基于一些仿真假设,通过块差错率(Block Error Ratio,BLER)达到10-2和10-3所需要的信噪比(Signal Noise Ratio,SNR)进行评判。
例如,仿真假设如表2所示。
表2
其中,AWGN表示加性高斯白噪声信道(Additive White Gaussian Noise,AWGN),QPSK表示正交相移键控(Quadrature Phase Shift Keyin,QPSK),LLR表示对数似然比(Likelihood Rate,LLR),CRC表示循环冗余校验(Cyclic Redundancy Check,CRC)。CApolar表示循环冗余校验辅助(CRC-aided,CA)polar码,是指利用CRC校验比特来选择最佳的路径。List-X是指译码时对多条路径进行扩展,最后在多条路径中选择最合适的路径作为译码结果。
根据待编码信息长度K来获取门限PerfThresh_K,K范围较小时PerfThresh_K=0.1dBK范围较大时,PerfThresh_K=0.3dB。根据译码列表L来获取门限PerfThresh_L,对于L=1,PerfThresh_L=0.4dB,对于L=2,PerfThresh_L=0.2dB,其它情况均为0.1dB。最后的判决门前PerfThresh=max(PerfThresh_K,PerfThresh_L)。
具体地,评判算法的过程如下。
(1)取序列A
与序列B在每一个仿真案例上进行比较,其中,对于每一个仿真案例:
如果A的性能比B差PerfThresh,FailCount_AB加1;
如果A的性能比B好PerfThresh,WinCount_AB加1。
如果(FailCount_AB–WinCountAB)/总仿真案例数量>2%,OverallFail_A加1。
与序列C在每一个仿真案例上进行比较,其中,对于每一个仿真案例:
如果A的性能比C差PerfThresh,FailCount_AC加1;
如果A的性能比C好PerfThresh,WinCount_AC加1。
如果(FailCount_AC–WinCountAC)/总仿真案例数量>2%,OverallFail_A加1。
对于序列D到N重复该过程。
(2)取序列B
与序列B在每一个仿真案例上进行比较,
依次类推。
……
(n)取序列N
与序列A进行比较,
以此类推。
……
选择OverallFail最小的序列。
如果多个序列A到M都有最小的OverallFail,
对于序列A,
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(性能差距超过PerfThresh),WinCount_AB加1。
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(性能差距超过PerfThresh),WinCount_AC加1。
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WinCount最高的序列,假定该序列为序列W。
如果某序列WinCount_xW>WinCount Wx,那么序列x也会被选择。
针对序列A,若wincount_AX相加的值越高(X为其他不同的序列),则序列A性能越好。
根据本申请的编码方法生成的排序序列获得的票数高于采用其它方法生成的排序序列获得的票数。
例如,采用本申请的编码方法生成的排序序列获得票数为255,而采用方案1的搜索算法生成的排序序列获得的票数为83,采用方案2的极化权重公式生成的排序序列获得票数为131,采用方案3的基于基序列和指示序列扩展得到的排序序列获得的票数为101、采用方案4的方法生成的排序序列获得票数为237。
由此可见,采用本申请实施例的极化编码方法生成的排序序列的性能有较大的提高。
方式2
可选地,第一序列可以通过如下方式生成:
根据至少一个第一指示序列对第一基础序列进行扩展,得到所述第一序列,其中,每个第一指示序列用于将一个长度等于n的序列扩展为长度等于2n的序列,第一基础序列包括S个极化信道的索引,该S个极化信道的索引是按照该S个极化信道的可靠度大小排序的。
在方式1中,通过采用第一序列替换最大母码长度对应的初始排序序列中的码长与第一序列相等的短排序序列,得到码长为N0的序列。其中,第一序列是采用方案4的方法生成的。
在方式2中,与方式1不同的是,采用方案3的方法生成第一序列。
根据前文对方案3的描述,可以通过对一个长度小于第一序列的短排序序列进行逐级扩展,直至达到第一序列的长度,扩展得到的序列即作为第一序列。
在本申请实施例中,将该用于扩展得到第一序列的短排序序列称作第一基础序列(或者,第一基序列)。
假定第一基础序列的长度等于R。换句话说,第一基础序列包括R个极化信道的索引,该R个极化信道的索引是按照该R个极化信道的可靠度的大小排序的。
具体地,对第一基础序列进行扩展时需要的第一指示序列的个数与第一基础序列的长度T和N0有关。
例如,N0=1024,T=512,则通过512扩展至1024的一个指示序列,即可将长度等于512的第一基础序列扩展至长度等于1024的序列。这个扩展过程仅需要1个指示序列。
又例如,N0=64,T=16,则先通过16扩展至32的一个指示序列,将长度等于16的第一基础序列扩展至长度等于32的序列,再通过32扩展至64的另一个指示序列,将该扩展后的序列再进一步扩展至长度等于64的序列。即,这个扩展过程需要2个指示序列。
若第一基础序列的长度等于R,需要最终扩展得到的序列的长度等于N0,则需要扩展的次数等于log2N0-log2R,需要的第一指示序列的个数也为log2N0-log2R。
需要说明的是,在本申请实施例中,将扩展过程中使用的所有指示序列都称作第一指示序列。当需要使用的第一指示序列的个数大于1个时,这多个第一指示序列是互不相同的,并且长度也各不相等。每个第一指示序列的长度都等于2的整数次幂。
这样,发送设备通过第一指示序列,对第一基础序列逐级扩展,直至扩展得到第一序列。再使用扩展得到的第一序列替换初始排序序列中长度等于第一序列的短排序序列,则得到最大码长等于N0的序列。
发送设备可以将该扩展得到的长度为N0的序列进行存储,以便于在后续进行极化编码时,可以直接从该长度等于N0的序列中读取所需母码长度对应的排序序列。
下面给出不同长度的第一指示序列的示例。
(1)从64扩展至128的指示序列。
11111101111111011111011110111101101110110110110101101010101101011100101010101001010010010010001001000010000100000100000000000000
(2)从128扩展至256的指示序列。
1111111011111111110111111111011111101111101111011110111101110111011011101101101101101101101101011010110101011010101010110101010111010100101010101010010101001010010100100100100100100100100010001001000010001000010000100000100000010000000001000000000000000000
(3)从256扩展至512的指示序列。
11111111011111111111111101111111111110111111111011111111011111101111110111111011110111110111101111011110111011110111011101110111011011101101110110110111011011011011011011011011010110110101101101011010101101011010101101010101101010101010101101010101010101011101010101010100101010101010101001010101001010100101001010100100101001010010010010100100100100100100100100010010010001001000100010010001000100001000100010000100001000010000010000100000010000001000000100000000100000000010000000000001000000000000000000000000
(4)从512扩展至1024的指示序列。
1111111110111111111111111111111101111111111111111011111111111110111111111110111111111101111111101111111101111111011111110111111011111101111110111110111111011111011110111110111101111011111011101111011110111101110111101110111101110111011101110111011101110111011011101110110111011011101101110110110110111011011011011011011101101101101011011011011011011011010110110101101101011011010110110101101011010110101011010110101101010110101010110101011010101011010101010101101010101010101010110101010101010101010101010101010111010101010101010101010101010100101010101010101010100101010101001010101010010101001010101001010010101001010010101001010010100100101001010010010100100101001001001010010010010010010010100100100010010010010010010010001001001000100100100010010001001000100010001001000100010001000100010001000010001000100001000100001000010000100010000010000100001000001000001000010000001000001000000100000010000001000000010000000100000000100000000100000000001000000000001000000000000010000000000000000010000000000000000000000000000000
与方式1类似,最大母码长度对应的初始排序序列采用方案2计算得到时,会涉及几种参数的取值。参见表3,表3是几种参数的取值与第一序列不同长度的几种组合。在每一种组合下,由第一基础序列逐级扩展得到第一序列,再对最大母码长度对应的排序序列进行替换,最终得到的长度为N0的序列也互不相同。下面分别给出。
表3
组合 | 参数 |
1 | a=1,b=0,c=1/4,N=128,N<sub>0</sub>=1024 |
2 | a=1,b=0,c=1/4,N=256,N<sub>0</sub>=1024 |
3 | a=1.07,b=0.5,c=1/4,N=128,N<sub>0</sub>=1024 |
4 | a=1.07,b=0.5,c=1/4,N=256,N<sub>0</sub>=1024 |
其中,表3中的N表示第一序列的长度。
以下,分别将表3中4种组合下生成的长度为N0的序列分别记作序列5、序列6、序列7和序列8。
序列5
{0,1,2,4,8,16,3,128,256,512,32,5,6,9,64,10,17,12,18,33,129,20,34,7,24,257,36,65,11,66,40,130,513,13,19,68,14,48,21,258,132,72,35,22,25,37,136,80,26,514,260,38,67,144,41,28,96,69,131,264,42,15,49,516,70,160,44,272,73,50,133,23,74,52,259,520,134,81,27,137,76,39,384,82,192,56,528,29,288,138,97,84,145,43,261,515,30,140,98,71,262,146,45,88,640,161,265,51,100,148,46,320,768,75,162,53,266,135,104,77,544,152,273,54,164,83,268,517,57,193,78,274,139,112,518,194,289,85,58,168,385,521,31,141,276,99,147,576,86,290,196,60,263,142,522,89,101,280,176,47,529,292,149,90,321,524,200,102,163,267,105,530,92,322,55,150,545,296,106,153,386,641,79,165,269,208,113,532,275,154,59,546,324,166,108,270,519,195,114,304,169,536,277,87,156,388,769,61,224,328,548,197,291,116,170,577,278,62,143,523,281,91,293,177,578,198,392,103,552,336,172,282,642,120,201,294,525,178,323,93,531,151,400,107,580,297,202,284,526,180,94,352,560,209,325,155,533,387,109,770,298,204,167,644,271,115,305,584,210,326,547,184,416,110,534,300,157,537,329,225,306,549,117,212,648,389,171,279,592,63,330,538,158,308,772,226,390,550,118,337,199,579,283,173,656,393,216,553,121,179,540,332,228,608,295,174,581,338,122,448,643,203,312,776,181,285,554,95,394,527,353,561,232,340,582,205,401,672,182,299,556,211,286,784,124,396,585,185,354,562,206,327,645,240,535,402,301,586,111,344,564,213,771,417,186,646,307,159,593,356,539,404,227,649,302,119,588,331,551,214,418,896,309,188,594,391,704,217,360,568,333,541,229,408,800,123,650,310,175,609,420,596,339,657,218,313,555,449,773,230,334,542,233,652,395,368,610,220,583,450,774,125,341,658,314,557,183,600,424,673,287,234,777,397,563,355,612,207,403,660,241,558,342,832,452,587,187,316,674,398,565,236,778,345,647,357,616,126,589,432,303,664,242,785,405,566,346,676,215,595,456,780,358,569,189,705,419,590,361,244,786,348,651,311,624,406,706,190,801,362,597,409,570,219,680,335,897,421,543,231,653,464,788,369,611,410,659,248,598,315,802,422,221,364,708,451,572,370,775,127,654,425,601,343,613,412,792,235,688,317,559,480,804,222,661,453,602,372,833,426,712,237,614,675,318,779,399,617,347,604,243,834,433,567,454,662,359,808,428,618,238,665,376,898,591,457,677,434,781,245,720,349,625,407,787,666,363,571,191,836,458,678,436,620,350,782,707,465,626,246,816,411,681,789,365,599,460,668,423,900,249,573,736,371,840,466,709,803,440,628,366,682,250,790,574,413,655,481,793,603,373,805,468,689,427,710,904,223,615,319,848,684,414,794,482,632,713,374,806,690,252,835,455,605,663,429,912,472,619,809,377,714,796,435,692,484,606,864,239,721,837,430,667,378,899,621,459,810,716,437,679,783,351,627,488,817,722,247,838,696,461,928,622,438,812,669,467,841,737,380,818,629,441,724,462,901,683,251,791,575,496,842,670,820,367,738,902,469,630,849,442,711,795,415,685,905,483,728,633,375,691,844,470,740,807,686,253,850,634,444,960,715,473,824,693,797,485,607,906,379,865,744,431,852,717,913,474,694,811,486,723,798,636,489,908,697,476,866,718,839,254,752,914,813,381,623,856,725,439,929,698,490,819,671,868,463,916,739,814,497,631,843,726,443,930,821,700,492,903,729,872,382,845,741,920,498,635,822,687,932,471,851,730,825,445,961,742,846,500,745,907,880,732,255,962,637,853,446,826,695,936,799,475,746,909,867,719,487,915,753,854,964,504,699,828,910,477,748,857,869,638,944,383,815,754,917,858,491,727,968,870,701,478,931,873,756,860,823,493,918,702,874,921,731,499,847,933,743,976,494,881,922,760,827,934,733,501,876,963,882,639,937,447,855,924,747,829,992,734,502,965,884,938,749,830,911,505,859,755,945,966,871,940,506,750,888,969,946,757,861,919,479,875,703,970,948,862,977,758,508,923,877,972,935,761,883,978,952,495,878,762,925,993,885,980,939,735,503,831,926,994,886,764,967,941,984,889,947,507,996,751,942,890,971,949,863,1000,759,973,950,509,979,892,953,974,763,1008,879,981,954,927,995,887,982,510,765,956,985,997,891,943,986,998,1001,988,766,893,951,1002,975,1009,955,511,1004,894,1010,983,957,1012,767,958,987,999,1016,989,895,1003,990,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列6
{0,1,2,4,8,16,3,32,256,512,5,6,9,64,10,17,12,18,128,33,20,34,7,24,36,257,65,11,66,40,13,19,513,68,14,129,48,21,72,258,130,35,22,25,132,37,80,26,38,514,260,67,136,41,28,96,69,42,15,264,144,49,70,516,44,73,131,272,50,23,74,160,133,52,259,520,81,27,76,134,39,82,288,137,56,29,528,192,261,97,138,84,43,30,262,515,145,98,71,140,265,45,88,146,544,51,320,100,46,75,161,266,768,148,53,104,273,77,162,135,517,54,268,83,152,57,274,518,78,164,193,384,112,139,521,85,289,58,31,194,276,576,99,168,290,86,141,522,60,263,89,147,280,196,529,101,142,292,47,90,524,321,176,149,267,530,102,200,105,322,92,640,163,296,150,55,545,269,153,106,275,532,79,165,324,208,113,546,270,154,59,519,385,108,166,304,536,195,114,277,169,548,328,87,156,386,769,61,224,291,577,197,278,170,116,523,281,143,62,578,388,91,293,177,552,198,336,103,525,282,172,294,531,201,120,323,580,93,392,178,526,151,297,202,641,284,107,352,560,94,180,325,533,209,298,155,584,271,204,770,400,109,167,642,305,210,326,547,115,300,534,184,329,157,537,387,110,644,306,225,279,549,212,171,592,330,117,416,538,158,389,550,63,308,772,226,337,579,199,283,118,648,332,173,553,390,216,540,121,179,608,295,228,581,338,174,554,393,203,312,527,122,285,776,95,448,656,181,353,561,394,582,232,340,556,205,299,585,124,286,643,182,401,784,211,354,562,185,327,535,206,396,586,111,672,301,344,645,240,402,564,213,771,307,186,593,356,159,539,302,588,331,227,646,417,214,551,404,594,119,309,800,360,188,649,333,568,217,418,541,391,704,229,310,773,175,609,339,218,650,408,596,313,555,123,334,542,420,774,230,449,657,233,610,368,395,583,220,652,341,777,125,314,557,287,600,183,658,450,563,355,234,832,424,612,342,558,207,397,587,126,673,316,778,345,660,241,403,565,452,616,187,785,357,589,398,674,236,647,303,346,566,242,780,432,595,405,664,215,569,358,786,456,590,189,676,361,705,348,244,896,419,624,406,651,311,597,190,801,409,570,362,543,219,706,335,788,421,611,231,680,464,802,369,598,410,653,248,572,315,775,364,221,422,601,451,659,370,792,235,708,425,613,343,654,412,804,222,559,317,602,480,833,127,688,453,661,426,779,372,614,237,712,617,399,834,318,604,347,675,243,808,433,662,454,567,359,781,428,665,238,618,457,787,591,376,677,349,836,245,720,434,625,407,782,666,458,571,191,897,363,620,350,678,436,816,707,465,626,246,789,411,681,840,460,599,365,668,423,898,249,573,736,466,803,371,709,790,440,682,413,628,250,793,655,366,574,481,900,603,468,805,427,689,373,710,848,223,615,414,794,684,319,806,482,713,632,455,835,605,252,904,374,690,663,429,809,472,714,796,377,619,864,435,606,484,692,837,239,721,810,430,667,459,783,716,378,912,621,351,679,817,437,722,488,838,627,247,812,696,461,841,669,380,899,622,438,818,737,467,791,724,441,683,462,842,629,251,928,670,367,901,575,820,496,738,849,469,711,795,442,630,844,415,685,902,483,728,633,470,691,807,375,740,850,686,253,905,715,444,824,634,473,797,607,960,485,693,865,431,906,744,474,852,717,811,379,636,798,486,694,913,723,489,866,697,476,839,718,908,254,623,813,856,381,752,914,725,439,819,698,490,868,671,814,463,843,739,929,382,726,916,631,497,821,872,700,443,845,729,930,492,903,741,822,498,687,851,730,920,471,825,635,846,445,932,742,961,500,745,880,907,732,255,853,637,826,446,799,695,962,867,475,746,936,854,719,487,909,638,828,857,504,753,915,964,477,699,869,910,748,815,491,858,754,944,917,478,727,870,968,701,383,873,860,756,931,918,493,823,702,921,874,731,499,847,933,743,976,494,881,922,760,827,876,733,501,934,963,882,747,937,447,855,924,734,829,992,639,502,965,938,884,749,911,830,505,859,755,945,966,871,940,506,750,969,888,861,757,946,919,479,970,703,875,862,948,977,758,508,923,972,877,935,761,978,883,952,495,925,762,878,993,980,939,885,735,503,926,831,994,967,764,886,941,984,889,947,507,996,751,942,971,890,863,949,1000,759,973,892,509,950,979,953,974,763,879,1008,981,954,927,995,982,887,510,765,956,985,997,943,986,891,998,1001,988,766,893,951,1002,975,894,1009,511,955,1004,1010,983,957,1012,767,958,987,999,1016,989,1003,990,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列7
{0,1,2,4,8,16,3,128,256,512,32,5,6,9,64,10,17,12,18,20,129,33,34,24,7,257,36,65,11,66,40,130,513,13,68,19,48,14,72,258,132,21,22,35,25,80,136,37,26,514,260,38,28,144,67,41,96,42,131,264,69,49,15,516,70,160,44,272,73,50,133,74,23,52,259,520,134,81,76,137,27,82,384,56,192,39,528,29,288,138,84,97,145,30,261,515,43,140,98,88,262,146,71,45,640,148,265,100,51,161,46,320,768,75,162,104,266,152,53,77,544,135,273,54,164,83,268,517,57,193,112,274,139,78,518,194,276,58,85,168,385,521,31,141,289,86,196,576,60,290,147,99,280,176,522,89,101,263,142,90,529,292,200,47,321,524,149,102,150,267,92,530,105,322,106,163,532,296,55,153,386,641,113,208,269,165,79,545,324,154,108,546,275,166,59,304,536,156,114,270,195,519,328,87,169,388,769,61,224,277,548,170,278,116,197,577,291,62,177,523,281,91,336,143,578,198,392,120,552,293,172,282,642,103,201,294,525,178,284,93,580,202,400,94,531,323,151,297,560,180,107,352,526,209,298,204,584,387,109,770,325,155,210,644,305,115,271,533,184,326,534,167,416,110,547,300,157,537,329,212,306,592,117,225,648,389,158,330,549,63,279,538,171,308,772,226,390,550,118,337,216,540,332,199,656,393,173,579,121,228,553,283,179,608,338,174,554,312,122,448,643,203,295,776,232,285,581,95,394,561,340,527,181,353,582,205,401,672,182,286,556,211,299,784,124,396,585,185,354,562,240,344,645,206,586,402,327,535,111,301,564,186,771,404,213,646,356,159,593,307,588,417,214,649,302,119,539,331,594,188,418,896,360,227,568,408,704,217,309,551,333,541,229,391,800,123,650,310,218,596,420,609,339,657,175,313,542,449,773,230,368,555,220,652,395,334,610,233,600,450,774,125,314,658,341,583,234,557,424,660,287,183,777,397,612,342,563,241,452,673,207,558,316,832,403,587,236,355,674,432,616,187,778,345,664,357,565,126,589,398,346,647,242,785,456,566,303,676,215,595,405,780,358,569,189,705,406,624,348,244,786,361,651,362,590,419,706,190,788,311,570,409,597,219,680,369,897,464,543,248,653,421,801,335,598,410,708,231,572,364,802,422,221,315,659,412,611,370,792,127,688,451,601,343,613,425,775,222,654,317,602,480,804,235,712,426,559,372,833,453,661,237,614,662,318,779,433,604,347,617,243,834,399,618,454,675,376,808,428,567,238,665,359,898,625,457,720,434,781,245,677,349,591,458,836,666,350,620,191,787,407,678,436,571,363,816,668,465,626,246,782,460,707,840,365,599,411,681,466,900,249,573,736,371,789,440,682,790,423,628,366,709,250,803,574,413,689,468,793,603,373,848,481,655,414,710,904,223,632,319,805,684,427,794,482,615,713,374,806,690,252,796,472,605,714,455,912,429,606,835,377,663,809,484,692,435,619,864,239,721,810,430,716,378,899,621,459,837,667,488,722,817,351,627,437,783,696,247,838,679,461,928,622,438,812,669,467,841,724,380,818,629,441,737,496,901,670,251,842,575,462,791,683,820,367,738,902,442,630,849,469,728,844,415,711,905,470,685,633,375,740,795,444,691,850,686,253,824,634,483,960,715,473,807,744,797,485,607,906,379,852,693,474,865,717,913,431,694,798,486,723,811,636,476,908,697,489,866,752,856,254,718,914,839,381,623,813,698,490,916,725,439,868,671,819,497,929,726,814,463,631,843,700,492,930,872,739,443,920,729,821,382,845,741,903,498,635,822,730,932,471,851,687,825,445,961,742,880,500,732,907,846,745,255,962,637,826,446,853,746,936,799,475,695,909,854,753,504,964,719,828,915,487,748,867,944,477,699,857,869,638,910,383,858,754,968,815,478,727,917,870,701,491,918,860,756,873,874,493,931,702,823,921,731,499,881,976,760,933,494,847,922,743,876,934,733,501,827,924,882,639,963,447,855,937,734,829,992,747,502,938,884,965,749,830,945,505,859,755,911,966,888,940,506,750,871,969,946,757,861,970,479,862,703,919,948,875,977,758,508,972,877,923,978,761,883,952,935,495,878,762,925,980,885,993,926,735,503,831,939,994,886,764,984,967,941,889,996,507,947,751,942,890,971,1000,863,949,759,973,950,509,979,892,953,1008,763,974,879,954,981,927,982,887,956,510,765,995,985,997,891,986,943,998,988,1001,766,893,1002,951,1009,975,1004,511,955,894,1010,983,957,1012,767,958,987,1016,999,989,895,990,1003,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列8
{0,1,2,4,8,16,3,32,256,512,5,6,9,64,10,17,12,18,128,20,33,34,24,7,36,257,65,11,66,40,13,68,513,19,48,14,129,72,21,258,130,22,35,132,25,80,37,26,136,514,260,38,28,67,41,96,42,144,69,264,49,15,70,516,44,73,50,272,131,160,74,23,52,133,259,520,81,76,134,27,82,56,288,137,39,192,528,29,261,84,97,138,30,43,262,515,98,145,140,88,265,146,71,45,544,100,320,51,46,148,161,266,768,75,104,53,273,162,152,77,517,54,268,135,83,164,274,518,57,112,78,384,193,58,521,85,276,139,194,168,289,576,31,86,290,60,99,522,141,280,196,89,263,147,529,176,101,292,142,90,524,321,47,200,267,530,102,149,92,322,105,640,150,296,163,106,532,269,153,55,324,545,208,165,275,113,154,546,304,79,108,536,270,166,59,385,519,114,156,328,195,548,277,169,224,386,769,87,61,278,577,116,291,170,197,523,388,62,177,578,281,143,336,198,552,91,293,172,525,282,120,392,580,201,178,294,531,103,284,93,560,202,323,151,526,297,180,352,641,94,107,298,584,209,400,204,533,325,155,770,305,210,184,642,271,109,326,534,167,300,547,115,329,110,644,306,157,537,387,212,416,592,225,117,549,330,158,279,538,171,308,648,226,389,772,63,337,550,216,332,118,540,390,199,579,283,173,553,228,121,608,338,179,554,312,174,656,393,122,295,581,203,448,776,232,285,561,181,340,527,353,582,95,394,556,124,286,585,205,299,562,182,354,784,211,401,643,185,396,672,240,344,586,206,535,402,327,564,186,301,645,111,771,356,213,593,307,159,588,302,646,404,214,539,417,188,594,331,568,227,360,800,309,217,649,418,551,119,408,704,333,541,229,310,773,218,596,391,175,609,339,650,420,542,230,313,555,368,774,123,334,610,220,657,449,314,652,233,600,341,777,125,395,658,450,583,234,557,424,612,287,183,832,342,563,316,558,126,355,660,241,673,397,778,452,587,207,345,616,403,565,236,785,432,674,357,664,187,589,398,346,566,242,780,303,647,456,595,215,676,358,786,405,569,189,624,348,590,361,244,896,406,705,419,570,362,597,190,788,409,651,311,706,219,680,464,801,421,543,248,598,369,802,410,572,335,611,231,653,364,792,422,221,315,708,370,601,412,775,222,659,451,688,425,613,480,804,235,654,343,602,317,833,127,559,372,712,426,779,453,614,237,661,604,318,834,433,617,399,662,243,808,454,675,347,618,428,781,376,665,238,567,457,836,720,434,677,359,787,245,625,349,666,458,816,591,407,620,191,782,436,678,350,571,363,897,626,465,668,246,840,460,707,789,411,681,466,736,440,898,249,599,573,365,790,423,628,803,371,682,366,709,250,900,574,413,689,468,793,655,481,848,373,710,414,603,805,223,684,427,794,632,482,904,319,713,690,472,806,615,252,796,374,605,714,455,835,429,663,809,484,692,864,377,606,435,619,810,239,721,912,430,716,378,837,667,459,817,722,488,696,783,437,621,351,838,679,247,812,627,380,841,622,461,818,669,438,899,724,467,928,737,441,629,496,842,670,251,791,683,462,820,738,901,442,575,849,367,728,844,469,630,902,415,711,795,470,685,740,444,633,850,483,691,824,686,253,905,634,473,807,715,375,960,744,797,485,693,852,474,865,607,431,906,636,798,486,717,811,379,694,866,723,476,913,697,489,908,752,856,254,718,914,839,381,698,813,623,490,868,725,439,819,671,814,382,916,726,929,497,700,843,739,463,872,821,729,492,930,631,920,443,845,741,822,498,730,903,687,851,471,932,742,825,445,880,635,846,500,732,961,826,745,255,853,637,907,446,962,746,936,799,475,695,854,828,638,504,867,753,909,964,487,719,857,915,477,748,944,869,699,910,478,858,754,815,968,491,727,870,917,701,383,860,873,756,918,931,493,874,702,921,823,731,499,976,933,760,881,494,922,847,743,876,934,733,501,827,882,924,734,963,447,937,992,747,855,829,639,502,884,938,965,749,830,945,505,911,755,966,859,940,888,506,750,969,946,871,757,861,970,479,919,703,948,862,875,977,758,508,972,923,978,952,761,877,935,883,495,878,762,925,980,993,885,926,735,503,939,994,831,984,764,886,967,941,996,889,507,947,751,942,890,971,1000,949,863,759,892,973,509,950,979,953,1008,763,974,954,879,981,927,982,956,995,510,765,985,887,997,986,943,998,891,988,1001,766,893,1002,951,894,1009,975,511,1004,955,1010,983,957,1012,767,958,987,1016,999,989,990,1003,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
另外,第一序列还可以采用方案1或方案2的方法生成。
方式3
可选地,最大母码长度对应的初始排序序列可以通过如下方式生成:
根据至少一个第二指示序列,对第二基础序列进行扩展得到最大母码长度对应的初始排序序列,其中,第二基础序列包括T个极化信道的索引,T个极化信道的索引是按照T个极化信道的可靠度的大小排序的,每个第二指示序列用于将一个长度等于n的序列扩展为长度等于2n的序列。
如前文所述,最大母码长度对应的初始排序序列可以采用以上方案1至4中的任意一种方法生成。
在方式3中,主要涉及采用方案3的方法生成最大母码长度对应的排序序列。即,通过对一个基础序列(即,第二基础序列)进行逐级扩展,最终扩展成为长度等于N0的序列。
这里,第二基础序列可以采用方案2的方法生成。例如,通过极化权重的公式(1)计算得到极化信道的可靠度,再根据极化信道的可靠度对极化信道的索引进行排序得到。可以理解的是,编号“第一”、“第二”仅仅是表达上的区分。因此,这里的第二基础序列与前文所述的第一基础序列可以相同或不同。实际上,不管是第一基础序列还是第二基础序列,都是作为一个基序列,最终扩展得到所需长度的排序序列。因此,第一基础序列和第二基础序列的生成方法、长度都不作特别限定,既可以相同,也可以不同。
另外,关于第二指示序列可以参见前文对第一指示序列所作的说明。例如,扩展次数、扩展过程中需要采用的第二指示序列的个数等都是类似的,这里不再赘述。
参见表4,表4是计算极化权重的参数与第二基础序列的不同长度的几种组合。在每一种组合下,由第二基础序列逐级扩展得到的长度为N0的序列也互不相同。下面分别给出。
表4
组合 | 参数 |
1 | a=1,b=0,c=1/4,T=256,N<sub>0</sub>=1024 |
2 | a=1,b=0,c=1/4,T=512,N<sub>0</sub>=1024 |
3 | a=1.07,b=0.5,c=1/4,T=256,N<sub>0</sub>=1024 |
4 | a=1.07,b=0.5,c=1/4,T=512,N<sub>0</sub>=1024 |
以下,分别将表4中4种组合下扩展得到的长度为N0的序列分别记作序列9、序列10、序列11和序列12。
序列9
{0,1,2,4,8,16,64,128,32,3,5,6,9,10,17,65,12,18,129,33,20,34,66,24,7,256,36,11,68,130,40,13,48,19,72,14,132,21,80,22,512,35,25,136,96,37,257,26,67,144,38,258,28,69,41,192,15,70,260,42,73,160,49,44,74,264,131,50,513,81,23,133,76,52,514,134,82,272,27,56,137,97,516,259,39,138,84,29,145,30,43,288,193,261,98,520,88,140,45,71,146,262,51,100,265,161,46,528,75,148,320,266,53,162,104,515,54,194,273,77,152,112,268,57,544,135,517,274,83,164,58,139,78,85,518,384,289,196,31,276,168,86,521,141,99,176,60,290,576,263,147,89,522,101,200,280,47,142,529,90,292,149,321,524,267,102,208,92,55,530,105,150,322,79,296,163,269,59,640,545,153,532,275,224,106,165,324,113,270,154,546,108,519,385,304,195,114,277,166,61,536,156,87,328,548,386,291,197,577,278,169,116,143,523,91,198,281,120,170,768,578,388,293,201,62,552,336,177,525,282,103,172,202,93,531,178,294,580,323,526,392,297,94,209,284,151,641,560,107,533,204,352,180,325,298,584,271,109,210,642,547,155,534,400,305,63,184,225,326,537,115,300,167,329,644,549,387,306,212,592,279,110,157,158,538,117,171,330,550,416,769,226,579,389,308,118,648,216,553,337,173,540,283,199,121,332,179,770,608,390,295,228,581,554,338,174,527,393,312,122,203,285,181,656,561,95,772,582,448,353,556,394,340,299,232,585,286,182,205,643,562,240,535,401,124,354,185,327,776,586,396,301,211,186,672,344,645,564,402,307,111,593,206,213,356,539,159,302,588,331,646,551,417,784,214,594,404,309,119,649,227,568,188,541,217,360,333,418,771,704,609,391,310,229,650,596,555,339,123,542,408,313,175,218,334,657,800,610,420,773,230,583,449,368,652,557,395,341,314,220,600,287,183,125,658,563,233,774,612,450,355,558,424,342,777,207,587,397,316,187,234,673,345,660,565,403,241,832,452,236,357,778,616,398,303,126,589,674,346,647,566,432,785,242,595,405,189,664,358,569,215,780,590,456,361,676,348,419,786,705,624,406,311,244,651,597,570,219,543,409,248,362,335,896,801,706,611,421,788,190,680,598,464,369,653,572,410,315,231,601,221,364,659,802,422,775,708,613,451,370,654,559,425,343,792,127,602,412,317,222,235,688,661,804,614,480,833,453,426,372,779,712,617,399,318,237,604,675,347,662,567,433,191,834,454,665,359,808,618,428,781,243,591,457,376,677,349,434,787,720,625,407,238,666,836,571,245,782,620,458,363,678,350,897,816,707,626,436,789,246,681,599,465,668,573,411,249,840,460,365,898,803,736,423,790,709,682,628,466,371,655,574,440,793,250,603,413,223,366,689,900,805,710,615,481,848,684,468,427,373,794,713,632,414,319,252,605,690,663,806,482,835,455,904,374,809,714,619,429,796,239,606,472,377,692,435,721,864,484,667,837,810,430,783,716,621,459,378,679,351,912,817,722,627,437,247,696,838,669,812,622,488,841,461,380,899,818,737,438,791,724,683,629,467,670,575,441,251,842,462,367,928,738,901,820,711,630,496,849,685,469,442,795,728,633,415,253,844,691,902,807,740,483,850,686,470,905,375,824,715,634,444,797,254,607,473,693,960,865,485,906,852,811,744,431,798,717,636,474,379,694,913,723,866,486,697,839,908,813,718,623,489,856,476,381,914,819,752,439,725,698,868,671,814,490,843,463,382,929,739,916,821,726,631,497,700,443,729,872,492,845,930,903,822,741,498,851,687,471,920,825,730,635,445,255,846,932,742,961,880,500,907,853,826,745,446,799,732,637,475,695,962,867,487,936,854,746,909,828,719,638,504,857,477,915,753,964,699,869,910,815,748,491,858,478,383,944,754,917,727,870,701,968,873,493,860,931,918,823,756,499,702,921,731,874,494,847,933,743,976,881,501,922,827,760,447,733,876,934,963,882,502,937,855,747,924,829,734,639,505,992,965,938,884,911,830,749,506,859,479,945,755,966,871,940,750,969,888,508,861,946,919,757,703,970,875,495,862,948,758,977,923,761,972,877,935,978,883,503,952,762,925,735,878,993,980,939,885,926,831,764,507,994,967,886,941,751,984,889,509,947,996,942,971,890,510,863,949,759,1000,973,892,950,979,953,763,974,879,1008,981,954,927,765,995,982,887,956,766,985,997,943,986,891,511,998,1001,988,893,951,1002,975,894,1009,955,1004,1010,983,957,767,1012,958,987,999,1016,989,1003,990,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列10
{0,1,2,4,8,16,64,128,256,32,3,5,6,9,10,17,65,12,18,129,33,20,34,66,257,24,7,36,11,68,130,40,13,48,19,72,14,258,132,21,512,80,22,35,25,136,96,37,260,26,67,144,38,28,69,41,192,264,15,70,42,73,160,49,272,44,74,131,513,50,81,23,320,133,514,76,52,134,82,27,384,56,516,137,97,39,288,138,84,29,145,30,259,43,193,520,98,88,261,140,45,71,146,262,51,100,161,528,46,265,75,148,53,266,162,515,104,54,194,273,77,152,112,321,544,57,517,135,83,268,164,58,139,274,518,78,85,196,385,31,168,289,521,86,141,99,276,176,576,60,290,147,522,89,101,322,200,47,529,280,142,90,263,524,149,102,208,292,92,530,55,267,105,150,324,79,163,386,640,545,59,532,153,296,224,106,269,165,113,304,546,154,519,108,275,195,114,328,166,270,536,61,156,388,548,87,197,277,577,169,336,116,143,523,278,91,198,291,120,768,578,281,170,201,392,552,62,352,525,177,293,103,172,282,531,202,323,580,93,526,178,400,94,294,209,284,641,560,151,533,107,325,204,297,180,584,448,109,271,642,547,210,534,155,326,63,298,184,329,537,225,416,115,305,644,549,167,300,212,592,330,110,157,387,538,158,306,117,550,337,769,171,579,279,226,389,648,118,553,332,216,540,308,173,390,199,121,770,608,179,338,228,581,554,283,174,527,312,122,393,203,353,181,656,561,295,772,582,95,394,556,340,232,285,182,585,401,205,286,643,562,240,535,299,124,449,185,354,776,586,211,344,186,396,672,301,645,564,111,327,206,593,402,213,307,539,159,356,588,417,646,551,214,784,302,594,119,331,227,649,404,568,309,541,188,418,217,360,771,704,609,310,229,450,650,596,555,123,333,542,175,408,368,218,313,657,800,610,391,773,230,583,339,220,652,557,420,314,183,395,600,125,334,341,658,563,233,774,612,452,287,558,207,424,777,342,587,187,397,234,355,673,432,660,565,241,316,832,403,236,345,778,616,357,126,456,589,674,303,647,566,242,785,398,595,346,189,664,405,569,358,780,590,215,464,676,348,244,786,705,624,311,361,406,651,597,570,219,543,335,419,248,315,896,801,706,611,409,788,190,680,598,480,362,653,572,421,231,369,601,410,221,659,802,364,775,708,613,451,370,654,559,127,422,792,317,602,412,222,343,453,688,661,804,614,235,833,425,372,399,779,712,617,237,347,454,604,675,376,662,567,191,426,834,457,665,318,808,618,433,781,243,591,359,428,677,458,238,787,720,625,349,434,666,836,571,350,782,620,245,465,678,407,897,816,707,626,363,789,460,681,599,246,668,573,436,365,840,466,411,898,803,736,249,790,709,682,628,319,440,655,574,481,793,371,603,250,423,468,689,900,805,710,615,366,848,684,413,414,223,794,713,632,373,427,482,605,690,663,806,374,835,252,904,472,809,714,619,429,796,455,606,377,435,692,484,721,864,239,667,837,810,430,783,716,621,378,459,679,437,912,817,722,627,351,488,696,838,669,812,622,247,841,438,461,899,818,737,496,791,724,683,629,380,670,575,441,467,842,251,442,928,738,901,820,711,630,367,849,685,462,469,795,728,633,415,470,844,691,902,807,740,375,850,686,483,905,253,824,715,634,444,797,473,607,485,693,960,865,379,906,852,811,744,431,798,717,636,474,486,694,913,723,866,476,697,839,908,813,718,623,439,856,254,381,914,819,752,489,725,698,868,671,814,463,843,443,490,929,739,916,821,726,631,497,700,492,729,872,382,845,930,903,822,741,498,851,687,445,920,825,730,635,471,500,846,932,742,961,880,255,907,853,826,745,475,799,732,637,504,695,962,867,446,936,854,746,909,828,719,638,487,857,477,915,753,964,699,869,910,815,748,383,858,478,491,944,754,917,727,870,701,968,873,493,860,931,918,823,756,447,702,921,731,874,499,847,933,743,976,881,494,922,827,760,501,733,876,934,963,882,502,937,855,747,924,829,734,639,505,992,965,938,884,911,830,749,506,859,479,945,755,966,871,940,750,969,888,508,861,946,919,757,703,970,875,495,862,948,758,977,923,761,972,877,935,978,883,503,952,762,925,735,878,993,980,939,885,926,831,764,507,994,967,886,941,751,984,889,509,947,996,942,971,890,510,863,949,759,1000,973,892,950,979,953,763,974,879,1008,981,954,927,765,995,982,887,956,766,985,997,943,986,891,511,998,1001,988,893,951,1002,975,894,1009,955,1004,1010,983,957,767,1012,958,987,999,1016,989,1003,990,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列11
{0,1,2,4,8,16,64,128,32,3,5,6,9,10,17,65,12,18,129,33,20,34,66,24,256,7,36,11,68,130,40,13,48,19,72,14,132,21,80,512,22,35,25,136,96,37,26,257,67,144,38,28,258,69,41,192,260,15,70,42,73,160,49,264,44,74,131,50,81,513,23,133,76,514,272,52,134,82,27,56,137,97,516,39,138,84,29,259,288,145,30,520,43,193,98,88,261,140,45,71,146,262,51,528,100,265,161,320,46,75,148,53,266,162,104,54,194,515,273,544,268,77,152,112,57,135,83,274,164,517,58,384,139,78,85,196,518,276,289,31,168,86,521,141,576,99,290,176,60,522,280,147,89,101,200,47,263,292,529,142,90,524,149,102,321,208,530,92,267,55,322,296,105,640,150,79,163,59,532,545,269,324,153,224,106,165,275,113,546,304,270,154,108,385,536,195,114,328,166,61,519,277,156,548,386,87,197,169,577,116,278,291,143,768,91,523,388,281,578,336,552,198,120,170,201,62,177,293,282,103,525,172,202,580,392,294,93,531,178,560,526,284,94,323,297,209,352,641,151,107,584,204,533,298,180,642,400,325,109,210,155,534,63,547,305,271,326,300,184,644,537,225,592,115,167,329,306,212,110,549,387,538,416,157,158,330,117,279,648,171,550,308,769,226,118,389,540,216,579,337,553,608,332,173,770,199,121,179,390,283,338,554,312,228,656,581,174,393,122,295,448,772,203,561,527,285,181,582,340,556,95,353,394,232,182,585,562,286,299,205,776,240,354,643,401,672,396,586,344,124,185,535,211,186,564,402,327,301,356,645,593,111,206,588,307,213,784,302,646,539,404,417,594,568,159,331,214,119,360,649,227,551,309,704,418,188,541,217,596,609,650,408,333,310,771,229,800,123,391,542,339,555,420,313,610,368,657,334,175,652,218,230,449,600,773,314,220,658,183,583,341,557,395,612,450,424,774,125,563,287,233,342,558,316,777,207,355,660,832,673,397,187,587,452,345,616,234,241,565,778,403,236,674,432,357,398,346,126,664,589,242,566,785,303,456,780,358,647,405,595,676,569,624,590,348,189,786,215,361,244,705,406,419,570,896,597,219,362,651,409,311,706,464,680,788,801,248,543,190,421,598,572,611,369,410,335,364,653,231,802,708,601,422,315,221,792,370,659,688,613,654,412,451,425,602,775,480,804,127,222,343,559,317,712,614,372,661,833,426,235,453,604,617,318,779,662,834,237,808,675,433,399,454,347,428,618,376,665,191,567,720,243,457,781,434,359,836,677,625,666,591,349,787,620,238,458,816,782,407,678,571,436,897,626,350,245,363,668,840,246,707,465,681,789,736,460,898,249,599,573,411,628,466,682,440,790,365,803,709,250,423,574,900,793,371,848,689,366,655,413,710,603,468,684,481,632,805,223,794,252,713,690,615,373,414,427,904,482,806,605,319,714,472,796,374,663,835,809,692,864,455,429,606,619,484,377,721,239,810,716,435,912,837,430,378,667,722,696,621,459,817,783,488,838,812,679,437,627,351,622,380,669,841,247,818,724,737,461,438,899,928,629,670,842,467,683,441,791,738,496,462,820,251,575,728,901,630,849,442,367,844,711,469,685,740,633,902,795,850,253,824,691,415,470,686,444,905,483,634,807,960,254,715,473,797,375,744,852,693,865,906,607,485,636,474,798,811,717,694,866,913,431,908,486,379,856,723,697,752,718,476,914,489,839,813,868,698,623,381,819,725,490,814,439,916,929,382,671,843,726,700,872,739,497,463,821,492,930,729,631,443,920,845,498,822,741,730,903,851,932,825,880,846,471,687,445,742,635,500,961,255,826,732,745,853,446,907,962,936,637,475,799,746,504,854,828,695,867,909,487,638,964,857,753,719,477,915,748,944,869,910,858,699,754,478,491,815,968,870,917,383,860,727,701,873,756,918,493,931,702,874,921,499,823,976,494,731,760,933,881,922,847,876,743,501,934,827,733,882,447,924,502,963,937,992,734,747,505,855,829,884,938,639,965,506,830,749,945,911,966,859,940,888,755,479,750,508,969,946,871,861,757,970,919,948,862,703,875,758,977,495,972,761,923,978,952,877,762,935,883,878,925,503,980,993,735,764,885,926,939,994,507,831,984,886,967,941,996,889,751,509,947,942,890,510,971,1000,949,863,892,759,973,950,979,953,1008,974,763,954,879,981,765,927,982,956,995,766,985,887,997,986,943,998,891,511,988,1001,893,1002,951,894,1009,975,1004,955,1010,983,957,1012,767,958,987,1016,999,989,990,1003,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列12
{0,1,2,4,8,16,64,128,256,32,3,5,6,9,10,17,65,12,18,129,33,20,34,66,257,24,7,36,11,68,130,40,13,48,19,72,14,258,132,512,21,80,22,35,25,136,96,37,260,26,67,144,38,28,69,41,192,264,15,70,42,73,160,49,272,44,74,131,50,513,81,23,320,514,133,76,52,134,82,27,384,56,516,137,97,39,288,138,84,29,145,520,30,259,43,193,98,88,261,140,45,71,146,528,262,51,100,161,46,265,75,148,53,266,162,104,54,515,194,544,273,77,152,112,321,57,135,83,268,517,164,58,139,274,78,85,518,196,385,31,168,289,521,86,576,141,99,276,176,522,60,290,147,89,101,322,200,47,529,280,142,524,90,263,149,102,530,208,292,92,55,267,105,640,150,324,79,163,532,545,386,59,153,296,224,106,269,165,546,113,304,154,108,275,536,195,114,328,166,270,519,61,156,548,388,87,197,277,577,169,336,116,143,768,278,523,91,198,578,291,552,120,281,170,201,392,62,352,177,293,525,103,172,580,282,202,323,531,93,560,526,178,400,94,294,209,284,641,151,107,584,325,533,204,297,642,180,448,109,271,210,534,155,547,326,63,298,184,329,644,537,225,592,416,115,305,167,300,212,549,330,538,110,157,387,158,306,117,648,337,550,171,769,279,226,389,540,118,579,332,553,608,216,308,770,173,390,199,121,179,338,554,228,283,656,581,174,312,122,393,203,772,353,561,527,181,295,582,95,556,394,340,232,285,182,585,562,401,205,286,776,240,299,643,124,672,449,586,185,354,211,535,344,186,564,396,301,111,327,645,593,206,402,588,213,307,784,159,646,539,356,417,594,568,214,302,119,331,227,649,404,551,309,704,188,418,541,217,596,609,650,360,310,229,771,450,800,123,333,542,175,555,408,368,610,218,657,313,391,652,230,339,220,600,773,420,314,658,183,583,395,557,125,612,334,341,774,233,563,452,287,207,558,424,777,342,187,660,832,673,397,234,587,355,432,616,241,316,565,778,403,236,674,345,357,126,456,303,664,589,242,566,785,398,346,780,189,647,405,595,676,569,624,590,358,215,786,464,348,244,705,311,361,570,896,597,406,219,651,335,419,706,248,680,788,801,315,543,409,190,598,572,611,480,362,421,231,653,369,802,708,601,410,221,364,792,451,659,688,613,654,370,127,422,602,775,317,804,412,222,343,559,453,712,614,235,661,833,425,372,399,604,617,237,779,662,834,347,808,675,454,376,191,426,457,618,318,665,433,567,720,243,359,781,428,458,836,677,625,666,591,238,787,620,349,434,816,782,350,678,571,245,897,626,465,407,363,668,840,460,707,246,681,789,736,436,898,365,599,573,466,628,411,682,249,790,319,803,709,440,481,574,900,793,371,848,689,250,655,423,710,603,468,684,366,632,805,413,794,414,713,690,615,223,373,427,904,482,806,605,374,714,252,796,472,663,835,809,692,864,429,455,606,619,377,435,721,484,810,716,239,912,837,430,378,667,722,696,621,459,817,783,437,838,812,679,351,627,488,622,247,669,841,438,818,724,737,461,496,899,928,629,670,842,380,683,441,791,738,467,251,820,442,575,728,901,630,849,367,462,844,711,469,685,740,633,902,795,850,415,824,691,470,375,686,483,905,253,634,807,960,444,715,473,797,485,744,852,693,865,906,607,379,636,431,798,811,717,694,866,913,474,908,486,476,856,723,697,752,718,439,914,254,839,813,868,698,623,381,819,725,489,814,463,916,929,443,671,843,726,700,872,739,490,497,821,492,930,729,631,382,920,845,498,822,741,730,903,851,932,825,880,846,445,687,471,742,635,500,961,255,826,732,745,853,475,907,962,936,637,504,799,746,446,854,828,695,867,909,487,638,964,857,753,719,477,915,748,944,869,910,858,699,754,383,478,815,968,870,917,491,860,727,701,873,756,918,493,931,702,874,921,447,823,976,499,731,760,933,881,922,847,876,743,494,934,827,733,882,501,924,502,963,937,992,734,747,505,855,829,884,938,639,965,506,830,749,945,911,966,859,940,888,755,479,750,508,969,946,871,861,757,970,919,948,862,703,875,758,977,495,972,761,923,978,952,877,762,935,883,878,925,503,980,993,735,764,885,926,939,994,507,831,984,886,967,941,996,889,751,509,947,942,890,510,971,1000,949,863,892,759,973,950,979,953,1008,974,763,954,879,981,765,927,982,956,995,766,985,887,997,986,943,998,891,511,988,1001,893,1002,951,894,1009,975,1004,955,1010,983,957,1012,767,958,987,1016,999,989,990,1003,895,1005,1011,1006,1013,959,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
方式4
可选地,第一基础序列可以通过如下方式生成:
获取该R个极化信道的极化权重,并将该R个极化信道的索引按照该R个极化信道的极化权重的大小进行排序,得到第一基础序列。
在方式4中,我们采用方案3生成前文所述的第一序列,并且用于扩展得到第一序列的第一基础序列是通过方案2生成。
即,根据公式(2)计算所述R个极化信道的极化权重,并根据极化权重的大小对该R个极化信道的索引进行排序,得到第一序列。最后,用第一序列替换最大母码长度对应的初始排序序列中码长等于第一序列的长度的短排序序列,生成长度等于N0的序列。
关于替换的过程前文已经作了详细描述,这里不再赘述。
在根据公式(1)计算这R个极化信道的极化权重时,参数(a,b,c)的取值可以为(1,0,1/4),(1.07,0.5,1/4),(1,0.72,1/4),(0.25,0.25,0.24)。
第一序列的长度N可以取值128、256或512。
参见表5,表5是4种不同参数的组合。根据每一组(a,b,c)的取值生成第一基础序列,再对长度等于R的第一基础序列经过逐级扩展,最终得到的长度等于N的第一序列。最后使用第一序列对最大母码长度对应的初始排序序列中长度等于N的短排序序列进行替换,得到长度为N0的序列。下面分别给出几种组合下得到的长度为N0的序列。
表5
以下,分别将表5中几种组合下生成的长度为N0的序列分别记作序列13、序列14、序列15和序列16。
序列13
{0,1,2,4,8,16,3,32,256,512,5,6,9,64,10,17,12,18,128,33,20,34,7,24,36,257,65,11,66,40,13,19,513,68,14,129,48,21,72,258,130,35,22,25,132,37,80,26,38,514,260,67,136,41,28,96,69,42,15,264,144,49,70,516,44,73,131,272,50,23,74,160,133,52,320,520,81,27,76,134,39,82,384,137,56,29,528,192,288,97,138,84,43,30,259,576,145,98,71,140,261,45,88,146,640,51,262,100,46,75,161,265,768,148,53,104,266,77,162,135,544,54,273,83,152,57,321,515,78,164,193,268,112,139,517,85,274,58,31,194,385,518,99,168,289,86,141,521,60,276,89,147,290,196,522,101,142,322,47,90,529,280,176,149,263,577,102,200,105,292,92,524,163,267,150,55,530,324,153,106,386,641,79,165,296,208,113,545,269,154,59,532,304,108,166,275,546,195,114,328,169,578,270,87,156,388,769,61,224,277,536,197,336,170,116,519,278,143,62,548,291,91,281,177,523,198,392,103,580,352,172,293,642,201,120,282,552,93,323,178,525,151,400,202,560,294,107,284,531,94,180,325,584,209,297,155,526,448,204,770,271,109,167,644,326,210,298,533,115,329,592,184,416,157,534,305,110,547,300,225,330,537,212,171,648,387,117,306,608,158,337,549,63,279,772,226,389,538,199,332,118,579,308,173,656,390,216,550,121,179,540,338,228,581,283,174,553,312,203,393,704,122,353,776,95,295,527,181,394,582,340,554,232,285,585,205,401,672,124,286,561,182,299,784,211,449,556,185,354,586,206,344,643,111,562,396,301,593,240,327,535,213,832,402,186,645,307,159,588,356,564,417,227,646,302,214,594,331,539,119,404,896,309,188,568,418,649,217,360,609,310,551,229,450,800,175,650,333,218,596,408,541,368,657,123,313,542,391,771,230,339,555,233,705,420,314,610,220,600,395,773,125,334,652,341,557,183,583,452,658,287,234,774,424,563,342,612,207,397,673,126,558,355,777,432,587,241,316,660,403,565,187,778,345,674,357,616,236,566,456,303,706,242,785,398,589,346,664,215,624,405,833,358,569,189,647,464,595,348,244,780,311,676,361,570,406,651,190,786,335,590,419,597,219,708,315,897,409,543,231,680,480,801,362,598,421,653,248,611,369,788,410,221,364,688,451,572,370,802,235,659,422,601,317,613,412,834,222,712,343,559,453,792,127,654,425,602,372,775,399,661,237,614,720,347,804,454,604,376,567,243,779,426,617,457,662,318,836,433,591,238,675,359,898,571,428,665,458,808,245,736,349,618,434,781,677,350,625,191,816,465,666,407,620,363,787,707,460,626,246,840,436,678,782,365,573,466,668,411,900,249,599,709,319,789,440,681,848,481,628,371,655,250,790,603,423,710,468,803,632,366,793,413,682,414,713,904,223,574,373,864,689,427,805,482,615,684,374,794,714,252,835,472,605,690,429,912,455,606,806,377,721,796,435,663,484,619,837,239,716,809,430,692,378,960,621,459,783,722,437,667,838,351,575,488,810,696,247,841,737,438,928,627,461,817,679,496,812,724,380,842,622,441,669,467,899,670,251,818,629,442,849,683,791,367,738,901,462,630,844,469,728,820,415,685,902,470,711,633,375,691,850,483,740,795,686,253,824,634,444,905,715,473,865,693,807,485,607,906,379,852,744,431,797,694,913,474,717,798,486,752,811,636,476,961,697,439,866,723,856,254,698,908,813,381,623,839,718,489,914,725,463,819,671,868,443,929,726,814,490,631,843,739,497,916,821,700,492,930,729,872,382,822,741,962,498,635,845,687,920,445,880,730,825,471,903,742,851,500,732,932,826,695,255,907,637,846,475,853,745,964,799,504,719,936,854,699,446,909,746,867,944,487,753,828,915,477,748,857,869,638,968,383,815,754,910,858,478,701,917,870,727,491,976,860,756,823,873,493,918,731,847,931,760,447,827,921,702,992,499,874,933,743,881,922,733,494,876,963,882,639,934,501,829,924,734,855,965,747,502,937,884,911,749,859,966,505,888,703,938,969,830,945,506,755,871,940,970,750,861,946,479,862,757,977,919,875,972,758,508,948,877,978,923,761,831,952,993,495,883,762,935,980,878,925,926,735,503,885,939,994,886,764,984,941,967,889,947,507,996,751,942,890,971,949,863,1000,759,950,973,509,1008,892,953,979,763,954,879,974,981,927,982,887,995,510,765,956,985,997,891,943,986,998,988,951,766,893,1001,975,955,1002,1009,511,1004,894,1010,957,983,1012,767,987,1016,958,999,989,895,990,1003,1005,959,1011,1006,1013,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列14
{0,1,2,4,8,16,3,32,5,512,6,9,64,10,17,12,18,128,33,20,34,7,24,36,65,11,256,66,40,13,19,68,513,14,129,48,21,72,130,35,22,25,132,37,80,26,38,257,67,514,136,41,28,258,96,69,42,15,144,49,260,70,44,516,73,131,50,23,264,74,160,133,52,81,27,520,76,134,39,272,82,137,56,29,259,192,528,97,138,84,43,30,145,288,98,576,261,71,140,45,88,146,51,262,640,100,46,265,75,161,148,53,768,320,266,104,77,162,135,54,544,273,83,152,57,268,78,515,164,274,193,112,139,85,517,58,31,384,289,194,99,518,276,168,86,141,60,521,89,147,290,263,196,101,522,142,47,280,90,176,529,149,292,102,321,577,267,200,105,92,163,524,150,55,322,153,530,296,106,269,79,641,165,275,208,113,154,545,324,59,270,532,108,166,385,304,546,195,114,277,169,578,87,156,61,328,769,386,291,224,536,278,197,170,116,519,143,62,281,548,91,177,388,293,523,198,103,336,580,172,282,201,642,120,93,178,552,151,294,323,525,392,297,202,560,107,284,94,531,180,209,352,584,155,325,298,526,271,204,770,109,167,400,644,305,210,115,533,184,326,592,157,300,110,534,329,387,547,306,225,279,537,212,171,648,117,158,63,608,330,416,549,226,389,772,308,199,538,118,337,173,579,283,216,656,121,332,550,179,390,540,295,228,581,338,174,553,393,312,203,704,122,285,776,95,181,527,448,353,582,394,554,340,299,585,232,286,672,205,124,561,182,401,784,211,354,556,185,327,586,396,301,643,206,562,111,344,593,402,307,535,240,832,213,186,645,356,159,588,302,564,331,417,646,227,404,594,309,539,214,119,896,188,217,568,360,649,333,418,609,391,551,310,229,800,339,650,175,408,596,313,541,218,657,123,334,542,420,771,230,449,555,368,705,395,341,610,314,600,233,773,287,220,652,125,557,183,583,450,658,355,424,774,342,563,234,612,397,316,673,207,558,126,777,345,587,403,241,660,452,565,187,778,357,674,398,616,303,566,236,346,706,432,785,242,589,405,664,215,624,358,833,189,569,456,647,361,595,348,419,780,406,676,311,570,244,651,190,786,409,590,219,597,362,708,335,897,421,543,231,680,464,801,369,598,410,653,315,611,248,788,221,364,422,688,451,572,370,802,425,659,343,601,235,613,412,834,317,712,222,559,127,792,480,654,453,602,426,775,372,661,399,614,720,318,804,237,604,347,567,433,779,243,617,454,662,359,836,428,591,238,675,457,898,571,376,665,349,808,434,736,407,618,245,781,677,191,625,458,816,363,666,350,620,436,787,707,246,626,465,840,411,678,782,249,573,460,668,365,900,423,599,709,466,789,371,681,848,440,628,250,655,413,790,603,223,710,366,803,632,481,793,468,682,427,713,904,373,574,414,864,689,319,805,252,615,684,482,794,714,455,835,374,605,690,429,912,239,606,806,472,721,796,377,663,435,619,837,484,716,809,430,692,459,960,621,378,783,722,351,667,838,437,575,247,810,696,488,841,737,461,928,627,380,817,679,438,812,724,467,842,622,441,669,251,899,670,462,818,629,367,849,683,791,496,738,901,469,630,844,442,728,820,415,685,902,253,711,633,483,691,850,470,740,795,686,375,824,634,444,905,715,254,865,693,807,473,607,906,485,852,744,431,797,694,913,474,717,798,379,752,811,636,486,961,697,489,866,723,856,476,698,908,813,381,623,839,718,439,914,725,490,819,671,868,463,929,726,814,382,631,843,739,497,916,821,700,443,930,729,872,492,822,741,962,498,635,845,687,920,471,880,730,825,445,903,742,851,255,732,932,826,695,500,907,637,846,446,853,745,964,799,475,719,936,854,699,487,909,746,867,944,504,753,828,915,477,748,857,869,638,968,491,815,754,910,858,478,701,917,870,727,383,976,860,756,823,873,493,918,731,847,931,760,499,827,921,702,992,494,874,933,743,881,922,733,501,876,963,882,639,934,447,829,924,734,855,965,747,502,937,884,911,749,859,966,505,888,703,938,969,830,945,506,755,871,940,970,750,861,946,479,862,757,977,919,875,972,758,508,948,877,978,923,761,831,952,993,495,883,762,935,980,878,925,926,735,503,885,939,994,886,764,984,941,967,889,947,507,996,751,942,890,971,949,863,1000,759,950,973,509,1008,892,953,979,763,954,879,974,981,927,982,887,995,510,765,956,985,997,891,943,986,998,988,951,766,893,1001,975,955,1002,1009,511,1004,894,1010,957,983,1012,767,987,1016,958,999,989,895,990,1003,1005,959,1011,1006,1013,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列15
{0,1,2,4,8,16,3,32,256,512,5,6,9,64,10,17,12,18,128,20,33,34,24,7,36,257,65,11,66,40,13,68,513,19,48,14,129,72,21,258,130,22,35,132,25,80,37,26,136,514,260,38,28,67,41,96,42,144,69,264,49,15,70,516,44,73,50,272,131,160,74,23,52,133,320,520,81,76,134,27,82,56,384,137,39,192,528,29,288,84,97,138,30,43,259,576,98,145,140,88,261,146,71,45,640,100,262,51,46,148,161,265,768,75,104,53,266,162,152,77,544,54,273,135,83,164,321,515,57,112,78,268,193,58,517,85,274,139,194,168,385,518,31,86,289,60,99,521,141,276,196,89,290,147,522,176,101,322,142,90,529,280,47,200,263,577,102,149,92,292,105,524,150,267,163,106,530,324,153,55,386,641,208,165,296,113,154,545,269,79,108,532,304,166,59,275,546,114,156,328,195,578,270,169,224,388,769,87,61,277,536,116,336,170,197,519,278,62,177,548,291,143,281,198,523,91,392,172,580,352,120,293,642,201,178,282,552,103,323,93,525,202,400,151,560,294,180,284,531,94,107,325,584,209,297,204,526,448,155,770,271,210,184,644,326,109,298,533,167,329,592,115,416,110,534,305,157,547,300,212,330,537,225,117,648,387,158,306,608,171,337,549,226,279,772,63,389,538,216,332,118,579,308,199,656,390,173,550,228,121,540,338,179,581,283,174,553,312,122,393,704,203,353,776,232,295,527,181,394,582,340,554,95,285,585,124,401,672,205,286,561,182,299,784,211,449,556,185,354,586,240,344,643,206,562,396,301,593,186,327,535,111,832,402,213,645,307,159,588,356,564,417,214,646,302,188,594,331,539,227,404,896,309,217,568,418,649,119,360,609,310,551,229,450,800,218,650,333,175,596,408,541,368,657,230,313,542,391,771,123,339,555,220,705,420,314,610,233,600,395,773,125,334,652,341,557,234,583,452,658,287,183,774,424,563,342,612,126,397,673,241,558,355,777,432,587,207,316,660,403,565,236,778,345,674,357,616,187,566,456,303,706,242,785,398,589,346,664,215,624,405,833,358,569,189,647,464,595,348,244,780,311,676,361,570,406,651,190,786,335,590,419,597,219,708,315,897,409,543,248,680,480,801,362,598,421,653,231,611,369,788,410,221,364,688,451,572,370,802,222,659,422,601,317,613,412,834,235,712,343,559,453,792,127,654,425,602,372,775,399,661,237,614,720,347,804,454,604,376,567,243,779,426,617,457,662,318,836,433,591,238,675,359,898,571,428,665,458,808,245,736,349,618,434,781,677,350,625,191,816,465,666,407,620,363,787,707,460,626,246,840,436,678,782,365,573,466,668,411,900,249,599,709,319,789,440,681,848,481,628,371,655,250,790,603,423,710,468,803,632,366,793,413,682,414,713,904,223,574,373,864,689,427,805,482,615,684,374,794,714,252,835,472,605,690,429,912,455,606,806,377,721,796,435,663,484,619,837,239,716,809,430,692,378,960,621,459,783,722,437,667,838,351,575,488,810,696,247,841,737,438,928,627,461,817,679,496,812,724,380,842,622,441,669,467,899,670,251,818,629,442,849,683,791,367,738,901,462,630,844,469,728,820,415,685,902,470,711,633,375,691,850,483,740,795,686,253,824,634,444,905,715,473,865,693,807,485,607,906,379,852,744,431,797,694,913,474,717,798,486,752,811,636,476,961,697,439,866,723,856,254,698,908,813,381,623,839,718,489,914,725,463,819,671,868,443,929,726,814,490,631,843,739,497,916,821,700,492,930,729,872,382,822,741,962,498,635,845,687,920,445,880,730,825,471,903,742,851,500,732,932,826,695,255,907,637,846,475,853,745,964,799,504,719,936,854,699,446,909,746,867,944,487,753,828,915,477,748,857,869,638,968,383,815,754,910,858,478,701,917,870,727,491,976,860,756,823,873,493,918,731,847,931,760,447,827,921,702,992,499,874,933,743,881,922,733,494,876,963,882,639,934,501,829,924,734,855,965,747,502,937,884,911,749,859,966,505,888,703,938,969,830,945,506,755,871,940,970,750,861,946,479,862,757,977,919,875,972,758,508,948,877,978,923,761,831,952,993,495,883,762,935,980,878,925,926,735,503,885,939,994,886,764,984,941,967,889,947,507,996,751,942,890,971,949,863,1000,759,950,973,509,1008,892,953,979,763,954,879,974,981,927,982,887,995,510,765,956,985,997,891,943,986,998,988,951,766,893,1001,975,955,1002,1009,511,1004,894,1010,957,983,1012,767,987,1016,958,999,989,895,990,1003,1005,959,1011,1006,1013,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
序列16
{0,1,2,4,8,16,3,32,5,512,6,9,64,10,17,12,18,128,20,33,34,24,7,36,65,256,11,66,40,13,68,19,513,48,14,129,72,21,130,22,35,132,25,80,37,26,136,38,257,514,28,67,41,96,258,42,144,69,260,49,15,70,44,516,73,50,264,131,160,74,23,52,133,81,76,520,272,134,27,82,56,137,39,192,29,84,528,97,138,259,288,30,43,98,145,576,140,88,261,146,71,45,100,262,640,51,46,265,148,320,161,75,768,104,53,266,162,152,77,54,544,273,268,135,83,164,57,515,112,78,274,193,58,384,517,85,139,194,168,276,289,518,31,86,60,99,141,521,290,196,89,280,147,176,522,101,142,90,263,292,529,47,200,102,149,577,321,92,105,267,150,524,322,296,163,106,530,153,55,208,269,641,324,165,113,154,79,545,275,108,304,532,270,166,59,385,546,114,156,328,195,578,169,277,224,386,769,87,61,116,536,170,278,291,197,519,62,388,281,548,336,177,143,198,523,91,172,120,580,293,282,201,642,178,103,392,552,294,93,202,525,284,151,323,560,297,180,352,531,94,107,209,584,298,204,400,526,325,155,770,210,184,109,644,305,271,326,533,300,167,592,115,110,157,534,329,306,547,212,225,387,537,416,117,648,158,330,171,608,279,226,549,308,63,772,216,389,538,118,337,332,579,199,173,656,228,121,550,390,283,540,338,312,581,179,174,553,393,122,295,704,448,203,776,285,232,527,340,181,582,353,554,394,95,585,124,286,672,299,205,561,182,354,784,401,396,556,344,211,586,185,240,643,206,562,402,327,593,301,356,535,186,832,111,307,645,213,302,588,404,564,417,159,646,331,214,594,188,539,360,227,896,309,418,568,217,649,119,408,609,333,551,310,229,800,218,650,391,339,596,420,541,313,657,368,334,542,175,771,230,123,555,449,705,314,220,610,233,600,341,773,395,450,652,424,557,125,583,287,658,234,342,774,316,563,183,612,355,397,673,126,558,452,777,345,587,241,207,660,403,565,236,778,432,674,357,616,398,566,346,187,706,242,785,303,589,456,664,358,624,405,833,348,569,215,647,189,595,361,244,780,406,676,419,570,190,651,362,786,409,590,311,597,464,708,219,897,248,543,421,680,369,801,410,598,335,653,364,611,231,788,422,315,221,688,370,572,412,802,451,659,425,601,480,613,222,834,235,712,343,559,317,792,372,654,426,602,127,775,453,661,318,614,720,237,804,433,604,399,567,454,779,347,617,428,662,376,836,243,591,238,675,457,898,571,434,665,359,808,349,736,245,618,458,781,677,407,625,436,816,350,666,191,620,363,787,707,246,626,465,840,460,678,782,249,573,411,668,466,900,440,599,709,365,789,250,681,848,423,628,371,655,366,790,603,413,710,468,803,632,481,793,223,682,252,713,904,373,574,414,864,689,427,805,482,615,684,319,794,714,472,835,374,605,690,455,912,429,606,806,484,721,796,377,663,239,619,837,435,716,809,430,692,378,960,621,459,783,722,488,667,838,437,575,351,810,696,380,841,737,247,928,627,461,817,679,438,812,724,467,842,622,441,669,496,899,670,462,818,629,251,849,683,791,442,738,901,367,630,844,469,728,820,253,685,902,415,711,633,470,691,850,444,740,795,686,483,824,634,254,905,715,473,865,693,807,375,607,906,485,852,744,474,797,694,913,431,717,798,486,752,811,636,379,961,697,476,866,723,856,489,698,908,813,381,623,839,718,490,914,725,439,819,671,868,382,929,726,814,497,631,843,739,463,916,821,700,492,930,729,872,443,822,741,962,498,635,845,687,920,471,880,730,825,445,903,742,851,500,732,932,826,695,255,907,637,846,446,853,745,964,799,475,719,936,854,699,504,909,746,867,944,487,753,828,915,477,748,857,869,638,968,478,815,754,910,858,491,701,917,870,727,383,976,860,756,823,873,493,918,731,847,931,760,499,827,921,702,992,494,874,933,743,881,922,733,501,876,963,882,639,934,447,829,924,734,855,965,747,502,937,884,911,749,859,966,505,888,703,938,969,830,945,506,755,871,940,970,750,861,946,479,862,757,977,919,875,972,758,508,948,877,978,923,761,831,952,993,495,883,762,935,980,878,925,926,735,503,885,939,994,886,764,984,941,967,889,947,507,996,751,942,890,971,949,863,1000,759,950,973,509,1008,892,953,979,763,954,879,974,981,927,982,887,995,510,765,956,985,997,891,943,986,998,988,951,766,893,1001,975,955,1002,1009,511,1004,894,1010,957,983,1012,767,987,1016,958,999,989,895,990,1003,1005,959,1011,1006,1013,1014,1017,1018,991,1020,1007,1015,1019,1021,1022,1023}
可以理解的是,根据前文所述的方案1至方案4描述的生成极化信道可靠度的排序序列的方法,可以有多种进行序列融合的方式,本申请实施例基于这样的思想,给出了上述4种方式的示例。
以上,对如何生成长度为N0的序列作了详细说明。发送设备生成该长度为N0的序列之后,可以对该长度为N0的序列进行存储。在进行极化编码时,根据信息比特的长度K,从该长度为N0的序列中读取所需长度的母码序列。
读取所需长度的母码序列之后,发送设备可以进一步根据该母码序列确定信息比特索引集合,从而对待编码比特进行极化编码,得到编码后的序列。
140、发送设备向接收设备发送编码后的序列。接收设备接收待译码序列。
在步骤140,发送设备将编码后的序列发送给接收设备,接收设备接收待译码序列。
这里,需要说明的是,发送设备对待编码比特进行极化编码,得到编码后的序列。该编码后的序列经过发送设备发送,被接收设备接收到的序列为待译码序列。
150、接收设备对待译码序列进行译码,得到译码后的序列。
接收设备对待译码序列进行译码的过程,正是发送设备对待编码比特进行极化编码的逆过程。
因此,与发送设备执行的操作或过程类似,接收设备也可以通过上述方式1至方式4中的方法,确定长度等于N0的序列。或者,也可以对该长度等于N0的序列进行存储。后续在进行译码时,可以从该长度等于N0的序列中直接读取所需长度的母码序列。进一步地,根据该母码序列确定信息比特索引集合。不失一般性地,接收设备从读取的母码序列中选取可靠度最高的前K个索引构成信息比特索引集合。
最后,接收设备根据信息比特索引集合对待译码序列进行译码,得到译码后的序列。
接收设备通过方式1至方式4确定长度为N0的序列的过程,可以参见前文对发送设备确定长度为N0的序列所作的说明,这里不再赘述。
在本申请实施例中,该长度为N0的序列还可以是极化信道的可靠度归一化序列。其中,该可靠度归一化序列也包括N0个极化信道的索引,与前述排序序列不同的是,可靠度的归一化序列中的每个索引对应一个极化信道的可靠度的量化值。
生成极化信道的可靠度归一化序列的过程如下:计算N0个极化信道的可靠度,将这N0个极化信道的可靠度进行量化(或者说,归一化),得到每个极化信道的可靠度的量化值。例如N0=1024,将这N0个量化值中的最大量化值赋值1023,再将次大的量化值赋值1022,依次类推,直至将最小的量化值赋值0。又例如,N0=128,将这N0个量化值中的最大量化值赋值127,再将次大的量化值赋值126,依次类推,直至将最小的量化值赋值0。将可靠度归一化序列记作Z()序列,则可以表示为:Z(Q(t))=t,其中,t为极化信道的索引,是指可靠度归一化序列的第Q(t)位的值为t,其中,t可以是1到N0,也可以是0到N0-1。
下面给出与序列1至序列16分别对应的可靠度归一化序列,并依次将其记作序列17至序列32。
序列17
{0,1,2,6,3,11,12,23,4,13,15,28,17,33,36,61,5,16,18,34,21,38,43,71,24,44,48,78,55,87,96,142,10,19,22,42,26,45,51,81,30,54,60,93,66,102,110,159,37,62,69,107,73,115,123,174,85,128,138,190,151,207,217,289,14,27,29,52,35,57,64,99,41,68,72,113,80,119,130,182,47,77,83,125,91,137,148,203,103,155,163,221,172,239,250,327,56,90,98,145,108,156,167,227,118,170,178,243,194,258,273,358,133,186,198,265,213,282,298,375,233,308,318,395,344,421,457,519,7,20,31,58,40,70,76,117,46,79,89,132,97,143,153,218,53,92,101,146,109,162,175,241,121,179,189,255,204,276,292,367,65,105,114,168,124,183,193,262,139,200,214,286,230,303,315,398,158,223,237,309,249,324,338,426,271,347,364,446,383,474,487,578,84,129,135,197,150,211,225,300,166,234,246,321,261,335,350,437,185,253,268,341,283,361,379,468,306,387,404,493,417,512,533,627,208,279,295,372,312,392,409,499,332,412,431,527,451,541,560,658,353,440,462,550,479,569,589,677,507,600,612,697,640,726,760,813,8,25,39,74,50,94,100,152,59,106,116,169,126,184,195,264,67,122,131,188,144,202,216,287,157,220,231,302,247,325,342,430,88,136,149,212,161,222,235,314,177,245,260,339,275,356,374,460,199,266,280,366,293,382,397,483,322,405,424,509,447,529,544,629,111,164,173,238,192,254,269,351,209,278,290,377,311,390,410,495,229,299,317,402,333,422,442,523,359,453,466,548,481,571,584,672,251,330,348,435,369,455,472,556,388,478,489,576,513,594,610,704,415,503,517,603,537,620,637,719,562,650,663,739,689,764,791,842,82,140,180,257,205,285,296,385,226,305,328,414,345,433,449,546,242,336,355,438,371,464,485,573,393,491,505,591,525,615,632,713,272,363,380,476,400,497,511,598,428,521,539,624,558,645,661,742,459,552,567,653,582,669,683,768,608,692,710,783,729,804,817,882,319,407,419,515,444,535,554,642,470,565,580,666,596,680,694,775,501,587,605,686,622,707,722,800,647,732,746,822,756,836,852,911,531,617,634,716,655,736,749,827,674,753,771,847,787,858,869,928,700,779,795,864,808,876,889,937,832,896,903,948,919,960,974,992,9,32,49,95,63,127,134,196,75,141,154,219,165,236,248,329,86,160,171,240,187,256,274,354,201,277,291,370,310,391,411,498,120,176,191,270,210,281,297,378,228,307,326,406,340,425,441,530,252,331,349,434,360,450,465,553,389,473,492,577,516,601,614,699,147,215,224,301,244,316,334,418,267,346,357,445,376,458,477,564,288,368,384,469,401,490,508,595,427,522,536,619,549,643,656,737,313,399,416,504,436,524,542,628,456,547,559,648,583,665,682,765,484,572,588,673,609,691,708,780,635,718,728,796,752,815,840,880,104,181,232,320,263,352,365,454,284,373,396,482,413,500,520,616,304,403,423,506,439,534,555,644,461,561,575,662,597,685,702,773,337,429,448,543,467,566,581,670,494,592,611,696,631,714,725,798,528,623,639,720,654,734,747,819,679,755,770,833,786,851,860,913,386,475,486,586,514,606,625,711,540,636,651,731,668,744,758,826,570,659,676,750,693,767,782,848,717,789,802,863,812,875,888,936,602,688,705,777,723,793,806,867,741,809,823,885,837,893,904,950,761,829,844,898,855,908,918,957,872,924,930,965,942,975,985,999,112,206,259,362,294,408,420,518,323,432,452,545,471,568,585,671,343,463,480,574,502,593,613,698,526,618,633,712,652,735,751,821,394,488,510,607,532,621,638,724,557,649,667,748,684,763,778,843,590,675,690,772,703,785,797,857,733,803,818,873,834,886,894,938,443,538,551,641,579,660,678,759,604,687,701,781,721,792,807,865,630,709,727,801,743,816,830,883,766,838,846,897,856,909,916,955,657,740,757,825,774,839,850,901,790,854,861,912,877,921,929,967,811,870,879,925,891,933,941,972,905,946,952,979,962,986,994,1005,381,496,563,664,599,695,706,788,626,715,738,810,754,824,835,895,646,745,762,828,776,845,859,910,794,862,871,920,884,931,939,970,681,769,784,853,799,866,874,923,820,881,892,935,902,944,951,980,841,899,907,947,915,954,959,987,927,963,969,991,976,997,1000,1012,730,805,814,878,831,890,900,943,849,906,914,953,922,958,964,989,868,917,926,961,934,968,973,996,945,977,981,1001,984,1004,1007,1016,887,932,940,971,949,978,982,1002,956,983,988,1006,993,1008,1010,1018,966,990,995,1009,998,1011,1013,1019,1003,1014,1015,1020,1017,1021,1022,1023}
序列18
{0,1,2,6,3,10,11,22,4,12,14,27,16,30,34,58,5,15,17,31,20,37,42,69,23,43,47,77,54,85,93,138,7,19,21,41,24,45,48,80,29,53,57,92,64,101,108,158,36,61,68,105,73,114,121,175,84,125,137,190,148,207,218,292,13,26,28,51,33,56,62,98,38,65,70,109,78,117,128,182,46,76,81,123,91,135,145,203,102,150,159,221,170,237,249,324,55,89,97,142,107,155,166,227,115,168,179,246,193,261,276,353,132,186,198,268,214,286,300,379,234,308,321,404,338,421,440,533,18,35,40,66,44,72,79,119,52,83,90,133,99,146,156,217,60,96,103,151,113,163,174,241,124,178,189,255,204,273,289,367,71,110,118,172,129,183,194,262,143,200,213,283,230,303,315,395,162,223,239,309,250,327,341,426,271,347,364,451,383,474,487,578,87,130,139,197,153,211,225,298,167,233,243,318,258,335,350,437,185,253,265,344,282,361,375,468,306,387,398,493,417,512,527,627,208,279,295,372,312,392,409,499,332,412,431,519,457,541,560,658,358,446,462,550,479,569,589,677,507,600,612,697,640,726,760,813,8,25,39,74,50,88,94,149,59,100,111,164,122,177,188,257,67,116,126,180,140,199,212,280,152,216,229,299,245,322,339,424,82,136,144,209,157,222,231,311,173,242,254,336,269,355,369,459,195,264,278,363,293,380,393,485,319,402,422,509,442,529,546,632,106,161,169,235,184,251,266,348,202,272,285,371,302,385,405,495,226,296,314,397,333,419,435,523,356,444,460,548,478,567,582,669,247,328,345,430,366,453,470,556,382,476,491,580,511,596,615,700,414,503,517,605,539,624,642,722,565,650,666,746,683,764,779,852,131,192,205,275,220,290,305,390,238,317,330,415,351,438,455,544,260,342,359,447,377,466,483,573,400,489,505,591,525,610,629,713,287,374,388,481,407,497,513,598,433,521,537,622,558,645,661,739,464,552,571,653,584,672,686,768,608,692,710,787,729,804,817,882,325,410,428,515,449,535,554,637,472,562,576,663,594,680,694,775,501,587,603,689,620,707,719,800,647,732,742,822,756,836,847,911,531,617,634,716,655,736,749,827,674,753,771,842,791,858,869,928,704,783,795,864,808,876,889,937,832,896,903,948,919,960,974,992,9,32,49,95,63,120,127,191,75,134,147,215,160,228,240,320,86,154,165,232,181,252,270,349,196,274,288,368,307,389,406,492,104,176,187,267,201,281,291,376,224,304,316,403,334,423,436,528,248,329,346,429,360,448,461,555,386,469,490,577,508,601,616,702,141,210,219,297,236,313,331,416,256,337,352,439,370,454,473,564,284,365,378,465,401,486,504,595,425,514,530,619,547,639,654,734,310,396,413,498,434,522,540,628,450,543,561,651,581,668,685,761,482,572,588,676,611,696,711,782,636,718,731,802,747,815,829,888,171,244,263,340,277,357,373,458,301,384,399,484,418,506,524,614,326,411,427,516,445,536,553,644,467,559,575,662,597,682,699,773,354,441,456,549,475,566,583,670,500,592,609,693,631,714,725,796,534,623,643,720,656,737,750,819,679,755,770,837,786,851,860,913,391,477,494,586,520,606,625,708,542,635,648,728,665,744,758,826,570,659,673,752,691,767,780,848,717,789,798,863,812,875,885,936,602,688,705,777,723,793,806,867,741,809,823,880,840,893,904,950,765,833,844,898,855,908,918,957,872,924,930,965,942,975,985,999,112,206,259,362,294,394,408,510,323,420,443,538,463,557,574,664,343,452,471,563,496,590,607,690,518,613,630,709,649,733,748,818,381,488,502,604,526,621,633,721,551,646,660,745,678,762,774,841,585,671,687,769,703,784,794,859,730,801,816,873,830,886,895,939,432,532,545,638,568,657,675,757,593,681,695,776,712,788,803,865,626,706,724,797,743,814,825,883,763,831,843,897,854,907,915,954,652,738,754,821,772,838,849,901,785,853,862,914,874,922,931,966,810,870,879,926,892,935,943,973,906,946,953,981,959,986,990,1007,480,579,599,684,618,701,715,792,641,727,740,811,759,828,839,894,667,751,766,834,781,846,857,910,799,861,871,920,884,929,938,970,698,778,790,856,805,866,877,923,824,881,891,934,902,944,951,979,845,899,909,947,916,955,961,987,927,963,969,993,976,997,1000,1012,735,807,820,878,835,890,900,941,850,905,912,952,921,958,964,989,868,917,925,962,933,968,972,996,945,977,980,1001,984,1004,1006,1016,887,932,940,971,949,978,982,1002,956,983,988,1005,994,1008,1010,1018,967,991,995,1009,998,1011,1013,1019,1003,1014,1015,1020,1017,1021,1022,1023}
序列19
{0,1,2,6,3,11,12,24,4,13,15,28,17,33,37,62,5,16,18,35,19,41,42,72,23,44,48,80,52,87,93,142,10,21,22,43,26,47,51,85,30,55,57,96,66,103,110,163,36,61,69,108,73,118,123,178,83,128,137,194,148,207,217,289,14,27,29,54,34,60,64,102,38,68,71,113,78,119,133,186,45,77,81,125,90,138,145,203,99,155,159,221,170,239,243,327,56,91,98,151,107,156,167,233,115,172,174,250,190,258,273,358,130,182,198,265,213,282,298,375,227,308,318,395,344,421,457,519,7,20,31,58,40,70,76,121,46,79,89,132,97,143,158,223,53,92,101,150,105,166,168,246,117,179,189,261,197,276,286,367,65,109,114,175,124,185,193,271,139,204,211,292,230,306,315,404,153,218,237,312,249,332,338,431,268,347,361,451,379,474,487,578,84,129,135,200,146,214,225,303,162,234,241,321,255,335,353,440,183,253,262,341,279,364,372,468,300,387,398,493,412,512,527,627,208,283,295,383,309,392,409,507,324,417,426,533,446,541,560,658,350,437,462,550,479,569,589,677,499,600,612,697,640,726,760,813,8,25,39,74,50,94,100,157,59,106,116,169,126,184,199,266,67,122,131,192,136,209,212,290,152,220,231,311,238,325,339,430,88,144,149,216,161,229,235,322,177,247,254,342,275,359,374,466,195,264,280,369,293,388,397,489,317,405,422,513,442,529,544,629,111,164,173,245,188,260,269,356,202,278,287,377,302,390,415,503,222,299,314,402,330,424,435,523,351,453,460,548,478,571,576,672,251,333,348,447,366,455,472,562,382,481,483,584,509,594,610,704,410,495,517,603,537,620,637,719,556,650,663,739,689,764,791,842,82,140,180,257,205,285,296,393,226,305,328,414,345,433,459,552,242,336,355,444,363,470,476,580,385,491,505,596,515,615,624,713,272,371,380,485,400,501,511,608,428,525,535,632,558,647,661,746,449,546,567,655,582,674,683,771,605,692,707,787,722,804,817,882,319,407,419,521,438,539,554,645,464,565,573,666,591,680,700,779,497,587,598,686,617,710,716,800,642,732,742,822,753,836,847,911,531,622,634,729,653,736,749,832,669,756,768,852,783,858,869,928,694,775,795,864,808,876,889,937,827,896,903,948,919,960,974,992,9,32,49,95,63,127,134,201,75,141,154,219,165,236,252,331,86,160,171,244,176,267,270,357,196,277,291,376,301,391,406,498,120,187,191,274,210,288,297,389,228,310,316,411,340,427,441,536,248,329,349,436,360,456,465,559,384,473,490,583,508,601,614,699,147,215,224,307,240,326,334,425,256,346,354,445,370,458,484,572,281,368,378,469,399,492,504,595,418,522,530,619,547,643,648,737,313,401,416,516,434,524,542,635,450,549,553,656,577,665,682,765,477,564,588,673,609,691,708,780,628,718,728,796,752,815,840,880,104,181,232,320,263,352,365,461,284,373,396,482,413,500,528,623,304,403,423,514,429,540,543,651,454,561,575,668,586,685,696,773,337,439,448,555,467,570,581,679,494,597,606,702,631,717,725,802,520,616,639,723,654,741,747,823,676,755,767,837,782,851,860,913,386,475,486,592,506,611,625,714,534,636,644,731,662,744,761,829,566,659,670,750,688,770,777,848,711,789,798,863,809,875,885,936,602,693,705,786,720,793,806,872,734,812,819,888,833,893,904,950,758,826,844,898,855,908,918,957,867,924,930,965,942,975,985,999,112,206,259,362,294,408,420,526,323,432,452,545,471,568,590,675,343,463,480,579,488,604,607,701,518,618,633,721,641,735,748,821,394,502,510,613,532,630,638,733,557,652,660,751,684,766,778,846,585,671,690,774,703,790,797,861,727,803,816,877,830,886,894,938,443,538,551,649,574,667,678,763,593,687,698,781,712,792,811,870,621,709,724,801,740,818,825,883,759,838,843,897,854,909,912,955,657,743,757,834,772,839,850,905,785,856,857,916,873,921,929,967,807,865,879,925,891,933,941,972,901,946,952,979,962,986,994,1005,381,496,563,664,599,695,706,794,626,715,738,810,754,824,841,899,646,745,762,831,769,849,853,914,788,862,871,922,878,931,935,970,681,776,784,859,799,868,874,927,820,884,890,939,902,945,951,981,835,895,907,949,915,956,959,988,926,963,968,993,973,997,1000,1012,730,805,814,881,828,892,900,944,845,906,910,953,920,958,966,990,866,917,923,961,932,969,971,996,943,977,980,1001,983,1004,1006,1016,887,934,940,976,947,978,982,1003,954,984,987,1007,991,1008,1010,1018,964,989,995,1009,998,1011,1013,1019,1002,1014,1015,1020,1017,1021,1022,1023}
序列20
{0,1,2,6,3,10,11,23,4,12,14,27,16,30,35,61,5,15,17,33,19,38,41,71,22,44,47,79,52,87,92,142,7,20,21,42,24,46,51,84,29,54,56,93,64,103,108,162,34,60,66,107,72,115,121,179,81,128,133,194,145,208,217,295,13,26,28,53,31,58,62,102,37,65,70,113,77,119,130,189,45,76,80,124,89,135,143,207,99,151,159,225,168,239,249,332,55,90,96,146,105,156,166,237,114,170,175,250,190,265,273,361,129,185,197,271,211,283,300,387,230,309,318,409,335,421,437,533,18,36,40,68,43,73,78,123,48,83,91,137,98,148,158,221,57,97,101,153,109,167,172,243,118,178,186,258,198,276,286,367,69,110,117,174,125,183,193,268,139,203,213,289,227,306,315,398,155,218,234,312,246,327,341,431,262,347,358,457,375,474,487,578,85,132,138,200,150,214,223,303,163,233,241,321,255,338,353,446,182,253,261,344,279,364,372,468,298,383,395,493,412,512,519,627,204,282,292,379,308,392,404,507,324,417,426,527,451,541,560,658,350,440,462,550,479,569,589,677,499,600,612,697,640,726,760,813,8,25,39,74,50,88,94,152,59,100,111,164,122,177,192,264,67,116,126,184,136,202,209,287,149,220,229,305,238,325,336,430,82,140,144,212,157,226,235,319,173,245,251,339,269,359,369,464,188,260,275,366,290,382,393,491,314,405,415,513,435,531,544,634,106,161,169,242,180,257,266,356,199,272,285,377,299,390,410,505,222,296,311,400,328,419,433,529,351,447,460,554,476,571,582,674,247,330,342,438,363,455,470,567,380,478,485,584,509,603,610,707,407,501,515,608,535,622,642,732,558,653,663,749,680,764,775,852,131,195,205,278,216,293,302,397,231,317,333,422,348,442,459,548,254,345,355,449,371,472,481,576,388,489,503,594,517,615,624,713,280,374,385,483,402,497,511,605,428,523,537,629,556,647,661,742,453,546,565,655,580,672,686,771,598,692,704,791,719,804,817,882,322,414,424,521,444,539,552,645,466,562,573,666,591,683,700,783,495,587,596,689,617,710,716,800,637,729,739,822,753,836,842,911,525,620,632,722,650,736,746,832,669,756,768,847,787,858,869,928,694,779,795,864,808,876,889,937,827,896,903,948,919,960,974,992,9,32,49,95,63,120,127,196,75,134,147,215,160,228,244,329,86,154,165,236,176,256,267,354,191,277,288,373,301,391,403,498,104,181,187,270,201,284,297,386,224,307,313,406,334,427,436,534,240,326,340,434,357,450,461,561,378,473,484,583,504,602,614,705,141,210,219,304,232,320,331,425,252,337,352,445,368,458,477,575,281,365,376,467,396,486,500,601,418,516,530,625,543,643,654,741,310,399,411,506,429,524,540,639,448,547,555,656,577,673,682,767,475,570,586,679,606,693,711,789,631,720,728,806,744,815,826,888,171,248,263,346,274,360,370,465,291,384,401,490,416,508,528,619,316,413,423,520,439,542,549,648,456,559,572,665,588,685,696,773,349,441,454,553,469,566,581,676,494,595,609,699,628,717,725,798,522,616,636,723,651,737,750,823,670,755,765,840,780,851,860,913,389,482,492,592,514,611,623,714,536,635,644,731,662,747,761,833,564,659,668,752,688,770,777,848,708,786,796,863,809,875,880,936,597,691,702,782,718,793,802,872,734,812,819,885,837,893,904,950,758,829,844,898,855,908,918,957,867,924,930,965,942,975,985,999,112,206,259,362,294,394,408,518,323,420,443,538,463,557,579,671,343,452,471,568,488,593,604,698,510,618,630,715,641,735,745,821,381,496,502,607,526,626,638,730,551,649,657,748,678,766,774,845,574,667,684,772,701,785,794,862,724,803,811,877,825,887,894,940,432,532,545,646,563,664,675,763,590,681,695,781,709,792,807,871,621,706,721,799,738,814,824,886,759,834,843,900,853,909,915,956,652,740,751,828,769,839,849,907,784,854,859,916,873,925,929,968,805,868,878,927,890,934,943,977,902,947,952,982,958,986,989,1007,480,585,599,687,613,703,712,797,633,727,743,816,757,830,841,897,660,754,762,835,776,850,856,912,790,861,870,921,879,931,935,970,690,778,788,857,801,866,874,926,820,883,891,938,901,945,951,980,838,895,906,949,914,955,961,988,923,963,967,994,972,997,1000,1012,733,810,818,881,831,892,899,944,846,905,910,953,920,959,966,991,865,917,922,962,932,969,971,996,941,976,979,1001,983,1004,1005,1016,884,933,939,973,946,978,981,1003,954,984,987,1006,993,1008,1010,1018,964,990,995,1009,998,1011,1013,1019,1002,1014,1015,1020,1017,1021,1022,1023}
序列21
{0,1,2,6,3,11,12,23,4,13,15,28,17,33,36,61,5,16,18,34,21,38,43,71,24,44,48,78,55,87,96,142,10,19,22,42,26,45,51,81,30,54,60,93,66,102,110,159,37,62,69,107,73,115,123,174,85,128,138,190,151,207,217,289,14,27,29,52,35,57,64,99,41,68,72,113,80,119,130,182,47,77,83,125,91,137,148,203,103,155,163,221,172,239,250,327,56,90,98,145,108,156,167,227,118,170,178,243,194,258,273,358,133,186,198,265,213,282,298,375,233,308,318,395,344,421,457,519,7,20,31,58,40,70,76,117,46,79,89,132,97,143,153,218,53,92,101,146,109,162,175,241,121,179,189,255,204,276,292,367,65,105,114,168,124,183,193,262,139,200,214,286,230,303,315,398,158,223,237,309,249,324,338,426,271,347,364,446,383,474,487,578,84,129,135,197,150,211,225,300,166,234,246,321,261,335,350,437,185,253,268,341,283,361,379,468,306,387,404,493,417,512,533,627,208,279,295,372,312,392,409,499,332,412,431,527,451,541,560,658,353,440,462,550,479,569,589,677,507,600,612,697,640,726,760,813,8,25,39,74,50,94,100,152,59,106,116,169,126,184,195,264,67,122,131,188,144,202,216,287,157,220,231,302,247,325,342,430,88,136,149,212,161,222,235,314,177,245,260,339,275,356,374,460,199,266,280,366,293,382,397,483,322,405,424,509,447,529,544,629,111,164,173,238,192,254,269,351,209,278,290,377,311,390,410,495,229,299,317,402,333,422,442,523,359,453,466,548,481,571,584,672,251,330,348,435,369,455,472,556,388,478,489,576,513,594,610,704,415,503,517,603,537,620,637,719,562,650,663,739,689,764,791,842,82,140,180,257,205,285,296,385,226,305,328,414,345,433,449,546,242,336,355,438,371,464,485,573,393,491,505,591,525,615,632,713,272,363,380,476,400,497,511,598,428,521,539,624,558,645,661,742,459,552,567,653,582,669,683,768,608,692,710,783,729,804,817,882,319,407,419,515,444,535,554,642,470,565,580,666,596,680,694,775,501,587,605,686,622,707,722,800,647,732,746,822,756,836,852,911,531,617,634,716,655,736,749,827,674,753,771,847,787,858,869,928,700,779,795,864,808,876,889,937,832,896,903,948,919,960,974,992,9,32,49,95,63,127,134,196,75,141,154,219,165,236,248,329,86,160,171,240,187,256,274,354,201,277,291,370,310,391,411,498,120,176,191,270,210,281,297,378,228,307,326,406,340,425,441,530,252,331,349,434,360,450,465,553,389,473,492,577,516,601,614,699,147,215,224,301,244,316,334,418,267,346,357,445,376,458,477,564,288,368,384,469,401,490,508,595,427,522,536,619,549,643,656,737,313,399,416,504,436,524,542,628,456,547,559,648,583,665,682,765,484,572,588,673,609,691,708,780,635,718,728,796,752,815,840,880,104,181,232,320,263,352,365,454,284,373,396,482,413,500,520,616,304,403,423,506,439,534,555,644,461,561,575,662,597,685,702,773,337,429,448,543,467,566,581,670,494,592,611,696,631,714,725,798,528,623,639,720,654,734,747,819,679,755,770,833,786,851,860,913,386,475,486,586,514,606,625,711,540,636,651,731,668,744,758,826,570,659,676,750,693,767,782,848,717,789,802,863,812,875,888,936,602,688,705,777,723,793,806,867,741,809,823,885,837,893,904,950,761,829,844,898,855,908,918,957,872,924,930,965,942,975,985,999,112,206,259,362,294,408,420,518,323,432,452,545,471,568,585,671,343,463,480,574,502,593,613,698,526,618,633,712,652,735,751,821,394,488,510,607,532,621,638,724,557,649,667,748,684,763,778,843,590,675,690,772,703,785,797,857,733,803,818,873,834,886,894,938,443,538,551,641,579,660,678,759,604,687,701,781,721,792,807,865,630,709,727,801,743,816,830,883,766,838,846,897,856,909,916,955,657,740,757,825,774,839,850,901,790,854,861,912,877,921,929,967,811,870,879,925,891,933,941,972,905,946,952,979,962,986,994,1005,381,496,563,664,599,695,706,788,626,715,738,810,754,824,835,895,646,745,762,828,776,845,859,910,794,862,871,920,884,931,939,970,681,769,784,853,799,866,874,923,820,881,892,935,902,944,951,980,841,899,907,947,915,954,959,987,927,963,969,991,976,997,1000,1012,730,805,814,878,831,890,900,943,849,906,914,953,922,958,964,989,868,917,926,961,934,968,973,996,945,977,981,1001,984,1004,1007,1016,887,932,940,971,949,978,982,1002,956,983,988,1006,993,1008,1010,1018,966,990,995,1009,998,1011,1013,1019,1003,1014,1015,1020,1017,1021,1022,1023}
序列22
{0,1,2,6,3,10,11,22,4,12,14,27,16,30,34,58,5,15,17,31,20,37,42,69,23,43,47,77,54,85,93,138,7,19,21,41,24,45,48,80,29,53,57,92,64,101,108,158,36,61,68,105,73,114,121,175,84,125,137,190,148,207,218,292,13,26,28,51,33,56,62,98,38,65,70,109,78,117,128,182,46,76,81,123,91,135,145,203,102,150,159,221,170,237,249,324,55,89,97,142,107,155,166,227,115,168,179,246,193,261,276,353,132,186,198,268,214,286,300,379,234,308,321,404,338,421,440,533,18,35,40,66,44,72,79,119,52,83,90,133,99,146,156,217,60,96,103,151,113,163,174,241,124,178,189,255,204,273,289,367,71,110,118,172,129,183,194,262,143,200,213,283,230,303,315,395,162,223,239,309,250,327,341,426,271,347,364,451,383,474,487,578,87,130,139,197,153,211,225,298,167,233,243,318,258,335,350,437,185,253,265,344,282,361,375,468,306,387,398,493,417,512,527,627,208,279,295,372,312,392,409,499,332,412,431,519,457,541,560,658,358,446,462,550,479,569,589,677,507,600,612,697,640,726,760,813,8,25,39,74,50,88,94,149,59,100,111,164,122,177,188,257,67,116,126,180,140,199,212,280,152,216,229,299,245,322,339,424,82,136,144,209,157,222,231,311,173,242,254,336,269,355,369,459,195,264,278,363,293,380,393,485,319,402,422,509,442,529,546,632,106,161,169,235,184,251,266,348,202,272,285,371,302,385,405,495,226,296,314,397,333,419,435,523,356,444,460,548,478,567,582,669,247,328,345,430,366,453,470,556,382,476,491,580,511,596,615,700,414,503,517,605,539,624,642,722,565,650,666,746,683,764,779,852,131,192,205,275,220,290,305,390,238,317,330,415,351,438,455,544,260,342,359,447,377,466,483,573,400,489,505,591,525,610,629,713,287,374,388,481,407,497,513,598,433,521,537,622,558,645,661,739,464,552,571,653,584,672,686,768,608,692,710,787,729,804,817,882,325,410,428,515,449,535,554,637,472,562,576,663,594,680,694,775,501,587,603,689,620,707,719,800,647,732,742,822,756,836,847,911,531,617,634,716,655,736,749,827,674,753,771,842,791,858,869,928,704,783,795,864,808,876,889,937,832,896,903,948,919,960,974,992,9,32,49,95,63,120,127,191,75,134,147,215,160,228,240,320,86,154,165,232,181,252,270,349,196,274,288,368,307,389,406,492,104,176,187,267,201,281,291,376,224,304,316,403,334,423,436,528,248,329,346,429,360,448,461,555,386,469,490,577,508,601,616,702,141,210,219,297,236,313,331,416,256,337,352,439,370,454,473,564,284,365,378,465,401,486,504,595,425,514,530,619,547,639,654,734,310,396,413,498,434,522,540,628,450,543,561,651,581,668,685,761,482,572,588,676,611,696,711,782,636,718,731,802,747,815,829,888,171,244,263,340,277,357,373,458,301,384,399,484,418,506,524,614,326,411,427,516,445,536,553,644,467,559,575,662,597,682,699,773,354,441,456,549,475,566,583,670,500,592,609,693,631,714,725,796,534,623,643,720,656,737,750,819,679,755,770,837,786,851,860,913,391,477,494,586,520,606,625,708,542,635,648,728,665,744,758,826,570,659,673,752,691,767,780,848,717,789,798,863,812,875,885,936,602,688,705,777,723,793,806,867,741,809,823,880,840,893,904,950,765,833,844,898,855,908,918,957,872,924,930,965,942,975,985,999,112,206,259,362,294,394,408,510,323,420,443,538,463,557,574,664,343,452,471,563,496,590,607,690,518,613,630,709,649,733,748,818,381,488,502,604,526,621,633,721,551,646,660,745,678,762,774,841,585,671,687,769,703,784,794,859,730,801,816,873,830,886,895,939,432,532,545,638,568,657,675,757,593,681,695,776,712,788,803,865,626,706,724,797,743,814,825,883,763,831,843,897,854,907,915,954,652,738,754,821,772,838,849,901,785,853,862,914,874,922,931,966,810,870,879,926,892,935,943,973,906,946,953,981,959,986,990,1007,480,579,599,684,618,701,715,792,641,727,740,811,759,828,839,894,667,751,766,834,781,846,857,910,799,861,871,920,884,929,938,970,698,778,790,856,805,866,877,923,824,881,891,934,902,944,951,979,845,899,909,947,916,955,961,987,927,963,969,993,976,997,1000,1012,735,807,820,878,835,890,900,941,850,905,912,952,921,958,964,989,868,917,925,962,933,968,972,996,945,977,980,1001,984,1004,1006,1016,887,932,940,971,949,978,982,1002,956,983,988,1005,994,1008,1010,1018,967,991,995,1009,998,1011,1013,1019,1003,1014,1015,1020,1017,1021,1022,1023}
序列23
{0,1,2,6,3,11,12,24,4,13,15,28,17,33,37,62,5,16,18,35,19,41,42,72,23,44,48,80,52,87,93,142,10,21,22,43,26,47,51,85,30,55,57,96,66,103,110,163,36,61,69,108,73,118,123,178,83,128,137,194,148,207,217,289,14,27,29,54,34,60,64,102,38,68,71,113,78,119,133,186,45,77,81,125,90,138,145,203,99,155,159,221,170,239,243,327,56,91,98,151,107,156,167,233,115,172,174,250,190,258,273,358,130,182,198,265,213,282,298,375,227,308,318,395,344,421,457,519,7,20,31,58,40,70,76,121,46,79,89,132,97,143,158,223,53,92,101,150,105,166,168,246,117,179,189,261,197,276,286,367,65,109,114,175,124,185,193,271,139,204,211,292,230,306,315,404,153,218,237,312,249,332,338,431,268,347,361,451,379,474,487,578,84,129,135,200,146,214,225,303,162,234,241,321,255,335,353,440,183,253,262,341,279,364,372,468,300,387,398,493,412,512,527,627,208,283,295,383,309,392,409,507,324,417,426,533,446,541,560,658,350,437,462,550,479,569,589,677,499,600,612,697,640,726,760,813,8,25,39,74,50,94,100,157,59,106,116,169,126,184,199,266,67,122,131,192,136,209,212,290,152,220,231,311,238,325,339,430,88,144,149,216,161,229,235,322,177,247,254,342,275,359,374,466,195,264,280,369,293,388,397,489,317,405,422,513,442,529,544,629,111,164,173,245,188,260,269,356,202,278,287,377,302,390,415,503,222,299,314,402,330,424,435,523,351,453,460,548,478,571,576,672,251,333,348,447,366,455,472,562,382,481,483,584,509,594,610,704,410,495,517,603,537,620,637,719,556,650,663,739,689,764,791,842,82,140,180,257,205,285,296,393,226,305,328,414,345,433,459,552,242,336,355,444,363,470,476,580,385,491,505,596,515,615,624,713,272,371,380,485,400,501,511,608,428,525,535,632,558,647,661,746,449,546,567,655,582,674,683,771,605,692,707,787,722,804,817,882,319,407,419,521,438,539,554,645,464,565,573,666,591,680,700,779,497,587,598,686,617,710,716,800,642,732,742,822,753,836,847,911,531,622,634,729,653,736,749,832,669,756,768,852,783,858,869,928,694,775,795,864,808,876,889,937,827,896,903,948,919,960,974,992,9,32,49,95,63,127,134,201,75,141,154,219,165,236,252,331,86,160,171,244,176,267,270,357,196,277,291,376,301,391,406,498,120,187,191,274,210,288,297,389,228,310,316,411,340,427,441,536,248,329,349,436,360,456,465,559,384,473,490,583,508,601,614,699,147,215,224,307,240,326,334,425,256,346,354,445,370,458,484,572,281,368,378,469,399,492,504,595,418,522,530,619,547,643,648,737,313,401,416,516,434,524,542,635,450,549,553,656,577,665,682,765,477,564,588,673,609,691,708,780,628,718,728,796,752,815,840,880,104,181,232,320,263,352,365,461,284,373,396,482,413,500,528,623,304,403,423,514,429,540,543,651,454,561,575,668,586,685,696,773,337,439,448,555,467,570,581,679,494,597,606,702,631,717,725,802,520,616,639,723,654,741,747,823,676,755,767,837,782,851,860,913,386,475,486,592,506,611,625,714,534,636,644,731,662,744,761,829,566,659,670,750,688,770,777,848,711,789,798,863,809,875,885,936,602,693,705,786,720,793,806,872,734,812,819,888,833,893,904,950,758,826,844,898,855,908,918,957,867,924,930,965,942,975,985,999,112,206,259,362,294,408,420,526,323,432,452,545,471,568,590,675,343,463,480,579,488,604,607,701,518,618,633,721,641,735,748,821,394,502,510,613,532,630,638,733,557,652,660,751,684,766,778,846,585,671,690,774,703,790,797,861,727,803,816,877,830,886,894,938,443,538,551,649,574,667,678,763,593,687,698,781,712,792,811,870,621,709,724,801,740,818,825,883,759,838,843,897,854,909,912,955,657,743,757,834,772,839,850,905,785,856,857,916,873,921,929,967,807,865,879,925,891,933,941,972,901,946,952,979,962,986,994,1005,381,496,563,664,599,695,706,794,626,715,738,810,754,824,841,899,646,745,762,831,769,849,853,914,788,862,871,922,878,931,935,970,681,776,784,859,799,868,874,927,820,884,890,939,902,945,951,981,835,895,907,949,915,956,959,988,926,963,968,993,973,997,1000,1012,730,805,814,881,828,892,900,944,845,906,910,953,920,958,966,990,866,917,923,961,932,969,971,996,943,977,980,1001,983,1004,1006,1016,887,934,940,976,947,978,982,1003,954,984,987,1007,991,1008,1010,1018,964,989,995,1009,998,1011,1013,1019,1002,1014,1015,1020,1017,1021,1022,1023}
序列24
{0,1,2,6,3,10,11,23,4,12,14,27,16,30,35,61,5,15,17,33,19,38,41,71,22,44,47,79,52,87,92,142,7,20,21,42,24,46,51,84,29,54,56,93,64,103,108,162,34,60,66,107,72,115,121,179,81,128,133,194,145,208,217,295,13,26,28,53,31,58,62,102,37,65,70,113,77,119,130,189,45,76,80,124,89,135,143,207,99,151,159,225,168,239,249,332,55,90,96,146,105,156,166,237,114,170,175,250,190,265,273,361,129,185,197,271,211,283,300,387,230,309,318,409,335,421,437,533,18,36,40,68,43,73,78,123,48,83,91,137,98,148,158,221,57,97,101,153,109,167,172,243,118,178,186,258,198,276,286,367,69,110,117,174,125,183,193,268,139,203,213,289,227,306,315,398,155,218,234,312,246,327,341,431,262,347,358,457,375,474,487,578,85,132,138,200,150,214,223,303,163,233,241,321,255,338,353,446,182,253,261,344,279,364,372,468,298,383,395,493,412,512,519,627,204,282,292,379,308,392,404,507,324,417,426,527,451,541,560,658,350,440,462,550,479,569,589,677,499,600,612,697,640,726,760,813,8,25,39,74,50,88,94,152,59,100,111,164,122,177,192,264,67,116,126,184,136,202,209,287,149,220,229,305,238,325,336,430,82,140,144,212,157,226,235,319,173,245,251,339,269,359,369,464,188,260,275,366,290,382,393,491,314,405,415,513,435,531,544,634,106,161,169,242,180,257,266,356,199,272,285,377,299,390,410,505,222,296,311,400,328,419,433,529,351,447,460,554,476,571,582,674,247,330,342,438,363,455,470,567,380,478,485,584,509,603,610,707,407,501,515,608,535,622,642,732,558,653,663,749,680,764,775,852,131,195,205,278,216,293,302,397,231,317,333,422,348,442,459,548,254,345,355,449,371,472,481,576,388,489,503,594,517,615,624,713,280,374,385,483,402,497,511,605,428,523,537,629,556,647,661,742,453,546,565,655,580,672,686,771,598,692,704,791,719,804,817,882,322,414,424,521,444,539,552,645,466,562,573,666,591,683,700,783,495,587,596,689,617,710,716,800,637,729,739,822,753,836,842,911,525,620,632,722,650,736,746,832,669,756,768,847,787,858,869,928,694,779,795,864,808,876,889,937,827,896,903,948,919,960,974,992,9,32,49,95,63,120,127,196,75,134,147,215,160,228,244,329,86,154,165,236,176,256,267,354,191,277,288,373,301,391,403,498,104,181,187,270,201,284,297,386,224,307,313,406,334,427,436,534,240,326,340,434,357,450,461,561,378,473,484,583,504,602,614,705,141,210,219,304,232,320,331,425,252,337,352,445,368,458,477,575,281,365,376,467,396,486,500,601,418,516,530,625,543,643,654,741,310,399,411,506,429,524,540,639,448,547,555,656,577,673,682,767,475,570,586,679,606,693,711,789,631,720,728,806,744,815,826,888,171,248,263,346,274,360,370,465,291,384,401,490,416,508,528,619,316,413,423,520,439,542,549,648,456,559,572,665,588,685,696,773,349,441,454,553,469,566,581,676,494,595,609,699,628,717,725,798,522,616,636,723,651,737,750,823,670,755,765,840,780,851,860,913,389,482,492,592,514,611,623,714,536,635,644,731,662,747,761,833,564,659,668,752,688,770,777,848,708,786,796,863,809,875,880,936,597,691,702,782,718,793,802,872,734,812,819,885,837,893,904,950,758,829,844,898,855,908,918,957,867,924,930,965,942,975,985,999,112,206,259,362,294,394,408,518,323,420,443,538,463,557,579,671,343,452,471,568,488,593,604,698,510,618,630,715,641,735,745,821,381,496,502,607,526,626,638,730,551,649,657,748,678,766,774,845,574,667,684,772,701,785,794,862,724,803,811,877,825,887,894,940,432,532,545,646,563,664,675,763,590,681,695,781,709,792,807,871,621,706,721,799,738,814,824,886,759,834,843,900,853,909,915,956,652,740,751,828,769,839,849,907,784,854,859,916,873,925,929,968,805,868,878,927,890,934,943,977,902,947,952,982,958,986,989,1007,480,585,599,687,613,703,712,797,633,727,743,816,757,830,841,897,660,754,762,835,776,850,856,912,790,861,870,921,879,931,935,970,690,778,788,857,801,866,874,926,820,883,891,938,901,945,951,980,838,895,906,949,914,955,961,988,923,963,967,994,972,997,1000,1012,733,810,818,881,831,892,899,944,846,905,910,953,920,959,966,991,865,917,922,962,932,969,971,996,941,976,979,1001,983,1004,1005,1016,884,933,939,973,946,978,981,1003,954,984,987,1006,993,1008,1010,1018,964,990,995,1009,998,1011,1013,1019,1002,1014,1015,1020,1017,1021,1022,1023}
序列25
{0,1,2,9,3,10,11,24,4,12,13,27,16,31,35,56,5,14,17,33,20,37,39,70,23,42,47,78,52,87,89,138,8,19,21,41,26,45,50,84,30,54,59,90,63,98,106,156,32,62,67,102,73,112,116,168,79,123,130,177,146,199,224,265,6,15,22,48,28,53,57,99,34,60,64,108,72,119,132,173,38,69,76,128,86,133,141,202,96,151,159,214,167,233,242,325,44,81,94,144,103,153,165,230,114,170,184,248,191,257,281,360,121,187,196,270,211,285,295,378,217,304,319,398,344,423,455,525,7,18,29,66,36,71,75,125,43,80,85,131,97,143,157,212,49,88,100,150,109,161,171,245,120,180,189,261,201,282,283,366,61,105,113,175,129,185,198,272,140,210,218,286,231,300,315,402,145,227,235,306,252,322,337,422,266,346,353,439,382,466,499,552,55,92,117,195,137,207,215,303,154,223,232,320,250,338,362,435,166,243,258,352,278,363,374,470,297,384,403,487,419,510,529,617,183,267,291,380,311,393,410,508,334,426,440,530,449,545,572,650,341,446,463,561,483,576,589,675,490,596,614,697,635,716,733,804,25,46,51,83,58,93,101,149,65,104,111,164,122,176,188,256,77,118,127,182,139,197,209,280,155,216,229,302,244,321,336,421,91,136,147,206,160,222,236,310,174,241,254,333,271,351,367,454,194,264,277,359,294,377,392,482,318,401,418,507,438,528,544,634,110,162,172,238,186,253,268,347,203,273,287,369,305,386,404,492,226,299,314,397,332,417,433,523,355,442,458,548,476,566,582,669,251,329,345,430,364,450,468,556,385,474,491,580,511,599,618,700,413,503,519,609,539,629,644,727,564,653,667,749,684,764,777,845,135,193,205,276,221,293,309,391,240,317,331,416,350,437,453,543,263,343,358,445,376,465,481,571,400,489,506,595,527,616,633,715,289,372,387,477,408,497,514,603,432,522,538,628,559,648,662,744,461,551,567,655,587,674,688,768,612,696,711,786,731,803,815,875,328,412,429,518,448,537,554,642,473,563,579,666,598,683,699,776,502,592,608,693,627,710,725,798,652,735,748,819,763,833,844,899,535,624,640,722,658,739,754,823,681,761,774,842,789,854,865,916,707,784,795,860,810,871,881,928,831,889,897,941,908,949,955,983,40,68,74,115,82,126,134,192,95,142,152,213,163,228,239,316,107,158,169,234,181,249,262,342,200,269,284,365,301,383,399,488,124,179,190,260,204,275,288,371,225,298,313,396,330,415,431,521,247,324,340,425,357,444,460,550,381,469,486,575,505,594,611,695,148,208,220,292,237,312,327,411,255,335,349,436,368,456,472,562,279,361,375,464,395,485,501,591,420,509,526,615,546,636,651,734,308,390,407,496,428,517,534,623,452,542,558,647,578,665,680,760,480,570,586,673,607,692,706,783,632,714,730,802,747,818,830,888,178,246,259,339,274,356,370,459,296,379,394,484,414,504,520,610,323,405,424,512,443,532,549,638,467,555,573,659,593,678,694,772,354,441,457,547,475,565,581,668,500,590,606,691,626,709,724,797,531,619,637,718,654,736,750,820,676,755,770,837,785,851,861,913,389,479,495,585,516,605,622,705,541,631,646,729,664,746,759,829,569,656,672,752,690,769,782,849,713,787,801,863,817,876,887,932,602,687,702,779,721,794,807,868,743,814,826,884,841,896,905,946,767,835,847,901,859,912,919,958,874,922,930,965,940,972,977,998,219,290,307,388,326,409,427,515,348,434,451,540,471,560,577,663,373,462,478,568,498,588,604,689,524,613,630,712,649,732,745,816,406,494,513,601,533,621,639,720,557,645,661,742,679,758,773,840,584,671,686,766,704,781,793,858,728,800,813,873,828,886,895,939,447,536,553,641,574,660,677,756,597,682,698,775,717,790,805,866,625,708,723,796,741,812,825,883,762,832,843,898,855,909,917,956,657,738,753,822,771,838,850,903,788,853,864,915,877,924,933,967,809,870,880,927,893,937,944,975,907,948,954,982,961,987,991,1007,493,583,600,685,620,703,719,792,643,726,740,811,757,827,839,894,670,751,765,834,780,848,857,911,799,862,872,921,885,931,938,971,701,778,791,856,806,867,878,925,824,882,892,936,904,945,952,980,846,900,910,950,918,957,962,988,929,964,970,993,976,997,1000,1012,737,808,821,879,836,891,902,943,852,906,914,953,923,960,966,990,869,920,926,963,935,969,974,996,947,978,981,1001,986,1004,1006,1016,890,934,942,973,951,979,984,1002,959,985,989,1005,994,1008,1010,1018,968,992,995,1009,999,1011,1013,1019,1003,1014,1015,1020,1017,1021,1022,1023}
序列26
{0,1,2,10,3,11,12,26,4,13,14,28,17,32,36,58,5,15,18,34,21,39,42,71,25,44,49,79,53,89,91,139,9,20,22,43,27,47,52,85,31,55,60,93,65,100,108,157,33,63,69,104,76,112,117,170,81,125,131,180,149,201,226,265,6,16,23,50,29,54,59,101,35,61,66,110,75,120,135,175,41,70,78,128,88,136,143,205,97,153,161,215,168,238,242,328,46,84,96,145,105,154,165,231,116,172,185,250,193,257,281,358,122,188,196,272,211,287,297,376,218,306,318,397,344,421,454,522,7,19,30,67,38,73,77,127,45,83,87,132,99,144,160,212,51,90,102,151,111,164,173,248,121,182,191,263,202,282,285,366,62,106,114,176,130,187,198,276,140,209,222,291,232,303,315,400,147,229,240,309,254,322,334,418,267,346,352,437,384,466,499,551,56,94,118,195,137,206,216,305,156,223,235,320,252,337,360,432,166,244,261,350,278,363,372,473,300,386,403,487,413,511,528,629,184,270,294,378,311,392,410,507,332,426,439,535,449,543,567,658,341,445,461,561,477,579,592,681,491,603,616,699,642,727,763,810,8,24,37,92,48,98,103,162,57,109,113,171,129,186,199,258,64,119,133,194,146,207,214,293,159,221,233,314,245,333,338,430,86,141,150,217,167,230,243,325,183,253,266,343,277,355,374,458,189,273,286,364,302,382,391,481,317,404,417,492,446,525,556,608,72,123,155,236,174,251,264,359,197,268,280,377,299,398,422,489,210,289,310,412,331,423,435,529,351,450,465,544,476,571,576,674,227,321,347,440,367,453,470,563,387,482,503,587,514,596,624,707,402,508,519,614,538,633,640,722,548,652,666,739,693,764,789,842,80,138,177,283,203,295,304,408,224,319,329,419,353,438,463,539,241,336,362,448,380,468,483,582,401,497,510,599,527,627,628,715,271,369,385,490,416,506,523,617,433,537,552,634,564,648,662,744,442,559,572,653,595,669,683,761,609,696,700,776,731,798,823,860,256,345,393,518,429,530,545,650,455,554,566,667,589,684,710,774,474,580,598,697,618,711,716,803,644,733,748,815,754,833,844,899,502,612,635,725,655,735,749,831,675,768,777,845,786,854,871,916,688,784,795,865,804,875,881,928,819,889,897,941,908,949,955,983,40,68,74,115,82,126,134,192,95,142,152,213,163,228,239,316,107,158,169,234,181,249,262,342,200,269,284,365,301,383,399,488,124,179,190,260,204,275,288,371,225,298,313,396,330,415,431,521,247,324,340,425,357,444,460,550,381,469,486,575,505,594,611,695,148,208,220,292,237,312,327,411,255,335,349,436,368,456,472,562,279,361,375,464,395,485,501,591,420,509,526,615,546,636,651,734,308,390,407,496,428,517,534,623,452,542,558,647,578,665,680,760,480,570,586,673,607,692,706,783,632,714,730,802,747,818,830,888,178,246,259,339,274,356,370,459,296,379,394,484,414,504,520,610,323,405,424,512,443,532,549,638,467,555,573,659,593,678,694,772,354,441,457,547,475,565,581,668,500,590,606,691,626,709,724,797,531,619,637,718,654,736,750,820,676,755,770,837,785,851,861,913,389,479,495,585,516,605,622,705,541,631,646,729,664,746,759,829,569,656,672,752,690,769,782,849,713,787,801,863,817,876,887,932,602,687,702,779,721,794,807,868,743,814,826,884,841,896,905,946,767,835,847,901,859,912,919,958,874,922,930,965,940,972,977,998,219,290,307,388,326,409,427,515,348,434,451,540,471,560,577,663,373,462,478,568,498,588,604,689,524,613,630,712,649,732,745,816,406,494,513,601,533,621,639,720,557,645,661,742,679,758,773,840,584,671,686,766,704,781,793,858,728,800,813,873,828,886,895,939,447,536,553,641,574,660,677,756,597,682,698,775,717,790,805,866,625,708,723,796,741,812,825,883,762,832,843,898,855,909,917,956,657,738,753,822,771,838,850,903,788,853,864,915,877,924,933,967,809,870,880,927,893,937,944,975,907,948,954,982,961,987,991,1007,493,583,600,685,620,703,719,792,643,726,740,811,757,827,839,894,670,751,765,834,780,848,857,911,799,862,872,921,885,931,938,971,701,778,791,856,806,867,878,925,824,882,892,936,904,945,952,980,846,900,910,950,918,957,962,988,929,964,970,993,976,997,1000,1012,737,808,821,879,836,891,902,943,852,906,914,953,923,960,966,990,869,920,926,963,935,969,974,996,947,978,981,1001,986,1004,1006,1016,890,934,942,973,951,979,984,1002,959,985,989,1005,994,1008,1010,1018,968,992,995,1009,999,1011,1013,1019,1003,1014,1015,1020,1017,1021,1022,1023}
序列27
{0,1,2,9,3,10,11,25,4,12,13,27,16,31,35,57,5,14,17,33,20,37,40,70,23,42,46,78,51,86,90,139,8,19,21,41,26,45,50,83,30,54,59,92,64,98,108,155,32,62,67,102,75,111,115,169,79,124,130,177,148,199,225,262,6,15,22,48,28,53,58,99,34,60,65,109,72,121,133,175,38,68,77,126,85,134,141,205,95,152,160,214,167,236,242,330,44,81,94,145,104,153,163,229,114,172,184,249,192,258,278,361,123,187,196,273,209,286,294,377,222,306,318,397,349,427,457,528,7,18,29,66,36,71,76,125,43,80,84,132,97,143,159,212,49,89,100,151,110,162,174,248,122,182,191,260,202,283,284,374,61,106,113,176,128,185,198,274,140,207,223,289,231,303,316,408,147,226,238,307,254,326,334,418,268,350,353,441,385,475,498,558,55,93,116,195,135,206,221,305,154,224,232,322,251,339,362,435,165,245,259,352,277,365,376,477,297,387,410,486,416,514,529,627,183,271,293,380,313,395,411,508,333,430,446,539,451,547,574,657,341,447,460,561,479,585,589,683,496,597,609,702,629,719,731,810,24,47,52,87,56,96,101,156,63,105,112,168,120,180,190,265,74,118,127,186,137,201,210,287,150,217,228,309,241,325,337,429,88,138,146,211,157,227,235,319,171,244,253,338,267,357,367,463,189,264,276,364,291,382,393,490,312,403,415,513,433,532,543,640,107,164,170,243,181,257,266,356,197,275,285,375,302,392,407,505,219,299,310,400,328,420,431,530,348,444,456,553,474,571,584,678,246,331,342,436,358,454,466,565,378,478,487,586,506,606,617,709,405,503,516,614,535,633,644,735,556,655,664,754,680,768,776,850,131,193,204,280,216,295,308,398,234,317,332,422,346,440,455,551,256,344,355,450,370,468,481,578,391,489,504,600,521,619,634,722,282,371,384,482,402,499,512,610,425,523,538,635,554,651,663,751,453,550,564,660,581,676,688,773,604,696,708,790,725,805,815,879,320,412,424,522,443,540,552,650,464,562,575,669,595,687,700,784,492,591,602,694,622,712,723,803,642,733,744,820,760,835,844,904,526,624,637,727,654,742,753,829,672,762,771,845,786,857,865,920,699,783,793,862,808,874,881,932,823,887,894,941,906,950,954,984,39,69,73,117,82,129,136,200,91,142,149,215,161,230,240,324,103,158,166,237,178,252,261,351,194,270,281,369,296,386,399,497,119,179,188,263,203,279,290,381,220,300,311,401,329,421,432,531,239,323,336,428,354,448,461,559,373,471,483,580,501,599,611,703,144,208,218,298,233,315,327,419,250,335,347,442,363,459,473,570,272,360,372,469,388,485,500,598,413,511,524,621,541,639,652,741,301,389,404,502,423,519,534,632,445,542,555,653,573,668,679,767,472,568,583,677,601,691,706,789,625,715,728,807,743,819,830,892,173,247,255,343,269,359,368,467,288,379,390,488,409,507,520,618,314,406,417,517,437,536,545,645,458,557,569,665,587,681,692,777,345,439,452,549,470,567,579,675,493,592,603,695,623,713,724,804,518,616,631,721,648,738,748,826,667,757,766,842,780,853,859,916,383,480,491,590,510,608,620,711,533,630,641,732,659,747,759,834,560,656,666,756,685,770,779,852,704,788,796,866,812,877,885,935,594,686,698,782,714,795,806,873,736,813,822,886,837,896,905,949,758,833,843,903,855,911,918,960,867,922,927,967,936,971,976,999,213,292,304,394,321,414,426,525,340,434,449,544,465,563,577,671,366,462,476,572,494,593,605,697,515,613,628,717,643,734,745,821,396,495,509,607,527,626,638,729,548,647,658,746,674,764,772,846,576,670,684,769,701,785,794,863,720,800,811,876,825,889,895,942,438,537,546,646,566,662,673,763,588,682,693,778,710,792,802,871,615,707,718,798,737,814,824,888,755,832,841,900,851,910,915,958,649,739,749,827,765,839,848,909,781,854,860,917,872,926,930,969,801,869,878,929,890,937,944,978,902,948,953,983,959,987,990,1007,484,582,596,689,612,705,716,797,636,726,740,816,752,828,840,898,661,750,761,836,774,849,856,913,791,861,870,923,880,931,938,972,690,775,787,858,799,868,875,928,818,883,891,939,901,946,952,981,838,897,908,951,914,957,962,989,925,964,968,994,974,997,1000,1012,730,809,817,882,831,893,899,945,847,907,912,955,921,961,966,992,864,919,924,963,933,970,973,996,943,977,980,1001,985,1004,1005,1016,884,934,940,975,947,979,982,1003,956,986,988,1006,993,1008,1010,1018,965,991,995,1009,998,1011,1013,1019,1002,1014,1015,1020,1017,1021,1022,1023}
序列28
{0,1,2,10,3,11,12,26,4,13,14,28,17,32,36,58,5,15,18,34,21,40,42,71,25,44,49,79,53,89,92,139,9,20,22,43,27,47,52,85,31,55,60,94,65,100,108,157,33,63,68,105,76,112,116,170,81,125,131,181,150,201,226,265,6,16,23,50,29,54,59,101,35,61,66,110,75,121,134,176,41,70,78,127,88,135,143,205,97,153,162,216,169,238,243,328,46,84,96,146,106,154,165,231,115,172,185,249,192,258,282,357,123,189,196,274,211,287,297,376,221,308,318,397,344,422,455,522,7,19,30,67,38,74,77,126,45,83,87,132,99,145,160,212,51,90,102,152,111,164,174,248,122,182,191,262,202,283,285,367,62,107,114,177,130,187,198,276,140,209,223,291,232,305,316,400,148,228,241,309,256,325,334,418,267,348,353,436,384,466,499,552,56,95,118,195,137,206,217,307,156,224,235,320,253,338,361,431,167,245,260,350,278,364,374,475,302,387,405,487,412,513,529,633,184,271,294,378,312,393,410,506,332,427,441,535,451,543,571,660,341,446,460,561,479,581,591,680,492,604,617,700,642,727,762,810,8,24,37,93,48,98,104,163,57,109,113,171,128,186,199,259,64,120,133,193,147,207,214,293,159,222,234,313,246,333,339,430,86,141,151,219,168,229,244,326,183,254,266,342,277,356,375,457,190,275,286,365,303,382,392,481,317,407,416,496,447,526,556,606,72,124,155,236,175,251,264,358,197,268,280,377,299,398,424,489,210,289,310,411,331,425,435,530,352,453,464,547,478,574,578,676,227,322,349,443,370,454,474,562,391,482,504,586,514,597,624,708,403,508,521,614,539,634,640,723,551,654,664,742,694,768,790,844,80,138,180,284,204,295,306,408,225,319,330,420,355,440,463,540,242,337,362,450,380,468,486,585,402,498,512,602,528,627,629,719,273,371,385,490,415,505,523,619,433,538,553,635,564,650,663,744,444,558,575,655,595,672,683,760,609,696,702,776,731,803,823,862,257,346,395,516,429,532,550,651,456,554,565,669,589,687,709,773,477,584,600,699,622,712,722,805,644,733,751,815,754,835,845,904,503,610,637,725,657,735,753,829,678,771,783,850,786,857,874,920,688,784,793,865,808,879,881,932,820,887,894,941,906,950,954,984,39,69,73,117,82,129,136,200,91,142,149,215,161,230,240,324,103,158,166,237,178,252,261,351,194,270,281,369,296,386,399,497,119,179,188,263,203,279,290,381,220,300,311,401,329,421,432,531,239,323,336,428,354,448,461,559,373,471,483,580,501,599,611,703,144,208,218,298,233,315,327,419,250,335,347,442,363,459,473,570,272,360,372,469,388,485,500,598,413,511,524,621,541,639,652,741,301,389,404,502,423,519,534,632,445,542,555,653,573,668,679,767,472,568,583,677,601,691,706,789,625,715,728,807,743,819,830,892,173,247,255,343,269,359,368,467,288,379,390,488,409,507,520,618,314,406,417,517,437,536,545,645,458,557,569,665,587,681,692,777,345,439,452,549,470,567,579,675,493,592,603,695,623,713,724,804,518,616,631,721,648,738,748,826,667,757,766,842,780,853,859,916,383,480,491,590,510,608,620,711,533,630,641,732,659,747,759,834,560,656,666,756,685,770,779,852,704,788,796,866,812,877,885,935,594,686,698,782,714,795,806,873,736,813,822,886,837,896,905,949,758,833,843,903,855,911,918,960,867,922,927,967,936,971,976,999,213,292,304,394,321,414,426,525,340,434,449,544,465,563,577,671,366,462,476,572,494,593,605,697,515,613,628,717,643,734,745,821,396,495,509,607,527,626,638,729,548,647,658,746,674,764,772,846,576,670,684,769,701,785,794,863,720,800,811,876,825,889,895,942,438,537,546,646,566,662,673,763,588,682,693,778,710,792,802,871,615,707,718,798,737,814,824,888,755,832,841,900,851,910,915,958,649,739,749,827,765,839,848,909,781,854,860,917,872,926,930,969,801,869,878,929,890,937,944,978,902,948,953,983,959,987,990,1007,484,582,596,689,612,705,716,797,636,726,740,816,752,828,840,898,661,750,761,836,774,849,856,913,791,861,870,923,880,931,938,972,690,775,787,858,799,868,875,928,818,883,891,939,901,946,952,981,838,897,908,951,914,957,962,989,925,964,968,994,974,997,1000,1012,730,809,817,882,831,893,899,945,847,907,912,955,921,961,966,992,864,919,924,963,933,970,973,996,943,977,980,1001,985,1004,1005,1016,884,934,940,975,947,979,982,1003,956,986,988,1006,993,1008,1010,1018,965,991,995,1009,998,1011,1013,1019,1002,1014,1015,1020,1017,1021,1022,1023}
序列29
{0,1,2,6,3,10,11,22,4,12,14,27,16,30,34,58,5,15,17,31,20,37,42,69,23,43,47,77,54,85,93,138,7,19,21,41,24,45,48,80,29,53,57,92,64,101,108,158,36,61,68,105,73,114,121,175,84,125,137,190,148,207,218,292,13,26,28,51,33,56,62,98,38,65,70,109,78,117,128,182,46,76,81,123,91,135,145,203,102,150,159,221,170,237,249,324,55,89,97,142,107,155,166,227,115,168,179,246,193,261,276,353,132,186,198,268,214,286,300,379,234,308,321,404,338,421,440,533,18,35,40,66,44,72,79,119,52,83,90,133,99,146,156,217,60,96,103,151,113,163,174,241,124,178,189,255,204,273,289,367,71,110,118,172,129,183,194,262,143,200,213,283,230,303,315,395,162,223,239,309,250,327,341,426,271,347,364,451,383,474,487,578,87,130,139,197,153,211,225,298,167,233,243,318,258,335,350,437,185,253,265,344,282,361,375,468,306,387,398,493,417,512,527,627,208,279,295,372,312,392,409,499,332,412,431,519,457,541,560,658,358,446,462,550,479,569,589,677,507,600,612,697,640,726,760,813,8,25,39,94,50,100,106,164,59,111,116,173,131,188,202,260,67,122,136,195,149,209,216,293,161,222,235,314,247,333,339,430,88,144,152,220,169,231,245,325,184,254,266,342,278,356,374,460,192,275,287,366,302,382,390,481,317,405,415,495,447,523,556,603,74,126,157,238,177,251,264,359,199,269,280,377,299,397,422,489,212,290,311,410,330,424,435,529,351,453,466,544,478,571,576,672,229,322,348,442,369,455,472,562,388,483,503,584,513,594,620,704,402,509,517,610,537,629,637,719,548,650,663,739,689,764,791,842,82,140,180,285,205,296,305,407,226,319,328,419,355,438,464,539,242,336,363,449,380,470,485,582,400,497,511,598,525,622,624,713,272,371,385,491,414,505,521,615,433,535,552,632,565,645,661,742,444,558,573,653,591,669,680,756,605,692,700,775,729,800,827,864,257,345,393,515,428,531,546,647,459,554,567,666,587,683,707,771,476,580,596,694,617,710,716,804,642,732,746,817,753,836,847,911,501,608,634,722,655,736,749,832,674,768,779,852,787,858,876,928,686,783,795,869,808,882,889,937,822,896,903,948,919,960,974,992,9,32,49,127,63,134,141,215,75,147,154,224,171,240,256,326,86,160,176,248,191,267,274,360,210,281,297,378,310,401,406,498,120,187,196,277,219,291,307,391,236,316,331,411,346,425,441,530,244,340,354,434,370,450,458,549,384,473,484,564,516,595,628,673,95,165,201,301,228,313,329,427,252,334,349,445,368,465,490,559,270,357,376,477,399,492,504,601,418,522,536,614,547,643,648,737,288,389,416,508,436,524,542,635,456,553,572,656,583,665,691,765,469,577,588,682,609,699,708,780,619,718,728,796,752,815,840,880,104,181,232,352,263,365,373,475,284,386,396,486,423,506,534,611,304,403,429,520,448,540,555,654,467,566,581,670,597,693,696,773,337,439,454,561,482,575,592,685,500,606,623,702,636,714,725,798,514,631,644,720,662,734,744,812,676,755,761,826,786,848,867,898,320,413,461,586,494,602,616,717,528,625,639,731,659,747,767,823,543,651,668,758,688,770,777,851,711,789,802,860,809,875,885,936,570,679,705,782,723,793,806,872,741,819,829,888,837,893,908,950,750,833,844,904,855,913,918,957,863,924,930,965,942,975,985,999,112,206,259,408,294,420,432,538,323,443,452,551,480,574,593,667,343,463,488,585,510,604,613,703,532,621,638,724,652,743,748,821,394,502,518,618,545,633,649,735,568,660,675,751,687,763,778,843,579,684,698,772,712,785,792,856,727,803,811,865,834,883,901,925,362,471,526,641,557,657,671,766,590,678,690,781,709,797,816,861,607,701,721,807,740,818,825,886,759,838,846,894,854,909,912,955,630,733,757,830,774,839,850,905,790,857,870,916,877,921,933,967,801,873,879,929,891,938,941,972,897,946,952,979,962,986,994,1005,381,496,563,695,599,706,715,805,626,730,738,814,762,828,845,892,646,745,769,835,784,849,859,915,799,866,874,923,884,934,935,970,681,776,788,862,810,871,881,931,824,890,899,939,906,944,951,980,831,902,910,947,920,954,958,984,926,963,966,989,976,996,1002,1009,664,754,794,878,820,887,895,945,841,900,907,953,917,959,968,988,853,914,922,964,932,969,971,997,943,977,981,1000,983,1004,1006,1016,868,927,940,973,949,978,982,1003,956,987,990,1007,993,1008,1011,1018,961,991,995,1010,998,1012,1013,1019,1001,1014,1015,1020,1017,1021,1022,1023}
序列30
{0,1,2,6,3,8,10,21,4,11,13,25,15,29,33,57,5,14,16,30,19,36,40,67,22,41,45,74,52,83,91,136,7,18,20,39,23,43,46,78,28,51,56,90,62,99,106,156,35,59,66,102,72,111,119,173,82,124,135,189,146,204,217,287,12,24,27,48,31,55,61,97,37,64,69,108,76,116,126,180,44,73,80,122,89,133,144,202,100,148,158,220,169,234,247,324,54,87,94,140,105,153,163,226,115,168,178,245,192,260,273,355,131,185,198,266,214,285,298,380,233,305,321,404,339,424,442,531,17,34,38,65,42,71,77,118,50,81,88,132,98,145,155,216,58,92,101,149,110,161,172,237,123,175,186,253,203,271,286,367,70,109,117,170,128,182,193,261,143,200,213,283,229,300,315,397,159,221,235,308,249,325,341,426,268,347,364,451,382,472,487,576,85,130,139,197,152,212,225,296,167,231,243,319,258,338,353,440,184,250,265,344,282,363,379,468,303,383,402,491,422,511,529,615,209,279,292,374,312,393,409,499,335,419,435,523,459,546,560,647,361,447,464,552,485,573,587,674,509,594,610,694,634,716,732,808,26,47,53,84,60,96,103,151,68,107,114,166,125,179,190,257,79,121,129,183,142,199,211,280,157,218,230,302,246,322,336,421,93,138,150,208,162,223,238,311,177,242,255,333,272,351,369,457,195,264,278,359,295,377,392,483,318,400,417,507,438,527,544,632,113,164,174,239,188,254,269,348,205,275,289,371,306,387,405,495,227,299,314,395,332,415,433,521,356,444,460,548,478,567,582,669,251,328,345,430,366,453,470,556,385,476,493,580,512,598,617,700,412,503,517,605,539,627,642,726,565,653,666,749,683,764,779,852,137,194,207,276,222,293,309,390,241,317,330,414,350,437,455,541,262,342,358,446,375,466,481,571,398,489,505,591,525,612,629,713,290,372,388,479,407,497,513,600,431,519,537,624,558,645,661,742,462,550,569,655,584,672,686,768,608,692,710,787,729,804,817,882,327,410,428,515,449,535,554,640,474,562,578,663,596,680,697,775,501,589,603,689,622,707,722,800,650,736,746,822,760,836,847,911,533,620,637,719,658,739,753,827,677,756,771,842,791,858,869,928,704,783,795,864,813,876,889,937,832,896,903,948,919,960,974,992,9,32,49,127,63,134,141,215,75,147,154,224,171,240,256,326,86,160,176,248,191,267,274,360,210,281,297,378,310,401,406,498,120,187,196,277,219,291,307,391,236,316,331,411,346,425,441,530,244,340,354,434,370,450,458,549,384,473,484,564,516,595,628,673,95,165,201,301,228,313,329,427,252,334,349,445,368,465,490,559,270,357,376,477,399,492,504,601,418,522,536,614,547,643,648,737,288,389,416,508,436,524,542,635,456,553,572,656,583,665,691,765,469,577,588,682,609,699,708,780,619,718,728,796,752,815,840,880,104,181,232,352,263,365,373,475,284,386,396,486,423,506,534,611,304,403,429,520,448,540,555,654,467,566,581,670,597,693,696,773,337,439,454,561,482,575,592,685,500,606,623,702,636,714,725,798,514,631,644,720,662,734,744,812,676,755,761,826,786,848,867,898,320,413,461,586,494,602,616,717,528,625,639,731,659,747,767,823,543,651,668,758,688,770,777,851,711,789,802,860,809,875,885,936,570,679,705,782,723,793,806,872,741,819,829,888,837,893,908,950,750,833,844,904,855,913,918,957,863,924,930,965,942,975,985,999,112,206,259,408,294,420,432,538,323,443,452,551,480,574,593,667,343,463,488,585,510,604,613,703,532,621,638,724,652,743,748,821,394,502,518,618,545,633,649,735,568,660,675,751,687,763,778,843,579,684,698,772,712,785,792,856,727,803,811,865,834,883,901,925,362,471,526,641,557,657,671,766,590,678,690,781,709,797,816,861,607,701,721,807,740,818,825,886,759,838,846,894,854,909,912,955,630,733,757,830,774,839,850,905,790,857,870,916,877,921,933,967,801,873,879,929,891,938,941,972,897,946,952,979,962,986,994,1005,381,496,563,695,599,706,715,805,626,730,738,814,762,828,845,892,646,745,769,835,784,849,859,915,799,866,874,923,884,934,935,970,681,776,788,862,810,871,881,931,824,890,899,939,906,944,951,980,831,902,910,947,920,954,958,984,926,963,966,989,976,996,1002,1009,664,754,794,878,820,887,895,945,841,900,907,953,917,959,968,988,853,914,922,964,932,969,971,997,943,977,981,1000,983,1004,1006,1016,868,927,940,973,949,978,982,1003,956,987,990,1007,993,1008,1011,1018,961,991,995,1010,998,1012,1013,1019,1001,1014,1015,1020,1017,1021,1022,1023}
序列31
{0,1,2,6,3,10,11,23,4,12,14,27,16,30,35,61,5,15,17,33,19,38,41,71,22,44,47,79,52,87,92,142,7,20,21,42,24,46,51,84,29,54,56,93,64,103,108,162,34,60,66,107,72,115,121,179,81,128,133,194,145,208,217,295,13,26,28,53,31,58,62,102,37,65,70,113,77,119,130,189,45,76,80,124,89,135,143,207,99,151,159,225,168,239,249,332,55,90,96,146,105,156,166,237,114,170,175,250,190,265,273,361,129,185,197,271,211,283,300,387,230,309,318,409,335,421,437,533,18,36,40,68,43,73,78,123,48,83,91,137,98,148,158,221,57,97,101,153,109,167,172,243,118,178,186,258,198,276,286,367,69,110,117,174,125,183,193,268,139,203,213,289,227,306,315,398,155,218,234,312,246,327,341,431,262,347,358,457,375,474,487,578,85,132,138,200,150,214,223,303,163,233,241,321,255,338,353,446,182,253,261,344,279,364,372,468,298,383,395,493,412,512,519,627,204,282,292,379,308,392,404,507,324,417,426,527,451,541,560,658,350,440,462,550,479,569,589,677,499,600,612,697,640,726,760,813,8,25,39,94,50,100,106,164,59,111,116,173,131,188,202,260,67,122,136,195,149,209,216,293,161,222,235,314,247,333,339,430,88,144,152,220,169,231,245,325,184,254,266,342,278,356,374,460,192,275,287,366,302,382,390,481,317,405,415,495,447,523,556,603,74,126,157,238,177,251,264,359,199,269,280,377,299,397,422,489,212,290,311,410,330,424,435,529,351,453,466,544,478,571,576,672,229,322,348,442,369,455,472,562,388,483,503,584,513,594,620,704,402,509,517,610,537,629,637,719,548,650,663,739,689,764,791,842,82,140,180,285,205,296,305,407,226,319,328,419,355,438,464,539,242,336,363,449,380,470,485,582,400,497,511,598,525,622,624,713,272,371,385,491,414,505,521,615,433,535,552,632,565,645,661,742,444,558,573,653,591,669,680,756,605,692,700,775,729,800,827,864,257,345,393,515,428,531,546,647,459,554,567,666,587,683,707,771,476,580,596,694,617,710,716,804,642,732,746,817,753,836,847,911,501,608,634,722,655,736,749,832,674,768,779,852,787,858,876,928,686,783,795,869,808,882,889,937,822,896,903,948,919,960,974,992,9,32,49,127,63,134,141,215,75,147,154,224,171,240,256,326,86,160,176,248,191,267,274,360,210,281,297,378,310,401,406,498,120,187,196,277,219,291,307,391,236,316,331,411,346,425,441,530,244,340,354,434,370,450,458,549,384,473,484,564,516,595,628,673,95,165,201,301,228,313,329,427,252,334,349,445,368,465,490,559,270,357,376,477,399,492,504,601,418,522,536,614,547,643,648,737,288,389,416,508,436,524,542,635,456,553,572,656,583,665,691,765,469,577,588,682,609,699,708,780,619,718,728,796,752,815,840,880,104,181,232,352,263,365,373,475,284,386,396,486,423,506,534,611,304,403,429,520,448,540,555,654,467,566,581,670,597,693,696,773,337,439,454,561,482,575,592,685,500,606,623,702,636,714,725,798,514,631,644,720,662,734,744,812,676,755,761,826,786,848,867,898,320,413,461,586,494,602,616,717,528,625,639,731,659,747,767,823,543,651,668,758,688,770,777,851,711,789,802,860,809,875,885,936,570,679,705,782,723,793,806,872,741,819,829,888,837,893,908,950,750,833,844,904,855,913,918,957,863,924,930,965,942,975,985,999,112,206,259,408,294,420,432,538,323,443,452,551,480,574,593,667,343,463,488,585,510,604,613,703,532,621,638,724,652,743,748,821,394,502,518,618,545,633,649,735,568,660,675,751,687,763,778,843,579,684,698,772,712,785,792,856,727,803,811,865,834,883,901,925,362,471,526,641,557,657,671,766,590,678,690,781,709,797,816,861,607,701,721,807,740,818,825,886,759,838,846,894,854,909,912,955,630,733,757,830,774,839,850,905,790,857,870,916,877,921,933,967,801,873,879,929,891,938,941,972,897,946,952,979,962,986,994,1005,381,496,563,695,599,706,715,805,626,730,738,814,762,828,845,892,646,745,769,835,784,849,859,915,799,866,874,923,884,934,935,970,681,776,788,862,810,871,881,931,824,890,899,939,906,944,951,980,831,902,910,947,920,954,958,984,926,963,966,989,976,996,1002,1009,664,754,794,878,820,887,895,945,841,900,907,953,917,959,968,988,853,914,922,964,932,969,971,997,943,977,981,1000,983,1004,1006,1016,868,927,940,973,949,978,982,1003,956,987,990,1007,993,1008,1011,1018,961,991,995,1010,998,1012,1013,1019,1001,1014,1015,1020,1017,1021,1022,1023}
序列32
{0,1,2,6,3,8,10,22,4,11,13,26,15,29,34,60,5,14,16,31,18,37,39,70,21,42,45,78,50,84,91,142,7,19,20,40,23,44,47,82,28,52,55,92,62,101,106,161,33,59,65,105,71,114,119,178,80,126,132,194,144,208,216,293,12,24,27,51,30,57,61,100,36,64,69,111,74,118,129,186,43,73,79,124,85,135,143,207,97,150,157,225,167,238,249,333,53,87,93,145,102,155,163,234,113,168,175,250,189,262,272,363,128,184,197,271,209,283,298,387,227,306,318,410,335,426,440,537,17,35,38,67,41,72,77,123,46,81,88,136,96,146,156,222,56,94,99,152,108,164,170,242,117,177,185,258,198,273,285,372,68,110,116,174,125,183,193,269,138,202,211,287,226,303,315,407,153,221,233,314,246,328,341,435,261,350,361,460,377,476,485,582,83,131,137,200,149,214,223,302,162,231,239,322,254,339,353,447,179,251,260,348,278,366,375,474,295,385,395,495,415,513,525,622,204,279,290,380,305,393,409,509,325,417,430,527,451,544,560,655,351,446,462,558,479,571,587,680,497,594,605,700,624,713,729,813,25,48,54,89,58,98,103,158,66,107,115,169,122,180,192,265,76,121,130,188,139,203,212,289,151,218,230,309,241,324,336,428,90,140,148,213,159,229,237,319,173,245,253,338,268,358,367,464,190,264,276,364,292,382,392,491,312,402,414,512,433,531,541,637,109,166,172,243,182,257,266,356,199,275,286,374,300,390,405,505,220,299,311,398,327,419,431,529,347,444,459,552,472,569,580,674,247,330,342,437,359,455,468,567,379,478,487,584,507,603,612,707,404,501,515,610,533,627,642,736,556,653,663,753,677,764,775,852,133,195,205,280,217,296,308,397,235,317,332,421,345,438,457,548,255,344,355,449,369,470,481,576,388,489,503,596,517,615,629,716,282,371,383,483,400,499,511,608,424,521,535,632,554,647,661,746,453,546,565,658,578,672,686,771,600,692,704,791,722,804,817,882,321,412,422,519,442,539,550,645,466,562,573,666,591,683,697,783,493,589,598,689,617,710,719,800,640,732,742,822,756,836,842,911,523,620,634,726,650,739,749,832,669,760,768,847,787,858,869,928,694,779,795,864,808,876,889,937,827,896,903,948,919,960,974,992,9,32,49,127,63,134,141,215,75,147,154,224,171,240,256,326,86,160,176,248,191,267,274,360,210,281,297,378,310,401,406,498,120,187,196,277,219,291,307,391,236,316,331,411,346,425,441,530,244,340,354,434,370,450,458,549,384,473,484,564,516,595,628,673,95,165,201,301,228,313,329,427,252,334,349,445,368,465,490,559,270,357,376,477,399,492,504,601,418,522,536,614,547,643,648,737,288,389,416,508,436,524,542,635,456,553,572,656,583,665,691,765,469,577,588,682,609,699,708,780,619,718,728,796,752,815,840,880,104,181,232,352,263,365,373,475,284,386,396,486,423,506,534,611,304,403,429,520,448,540,555,654,467,566,581,670,597,693,696,773,337,439,454,561,482,575,592,685,500,606,623,702,636,714,725,798,514,631,644,720,662,734,744,812,676,755,761,826,786,848,867,898,320,413,461,586,494,602,616,717,528,625,639,731,659,747,767,823,543,651,668,758,688,770,777,851,711,789,802,860,809,875,885,936,570,679,705,782,723,793,806,872,741,819,829,888,837,893,908,950,750,833,844,904,855,913,918,957,863,924,930,965,942,975,985,999,112,206,259,408,294,420,432,538,323,443,452,551,480,574,593,667,343,463,488,585,510,604,613,703,532,621,638,724,652,743,748,821,394,502,518,618,545,633,649,735,568,660,675,751,687,763,778,843,579,684,698,772,712,785,792,856,727,803,811,865,834,883,901,925,362,471,526,641,557,657,671,766,590,678,690,781,709,797,816,861,607,701,721,807,740,818,825,886,759,838,846,894,854,909,912,955,630,733,757,830,774,839,850,905,790,857,870,916,877,921,933,967,801,873,879,929,891,938,941,972,897,946,952,979,962,986,994,1005,381,496,563,695,599,706,715,805,626,730,738,814,762,828,845,892,646,745,769,835,784,849,859,915,799,866,874,923,884,934,935,970,681,776,788,862,810,871,881,931,824,890,899,939,906,944,951,980,831,902,910,947,920,954,958,984,926,963,966,989,976,996,1002,1009,664,754,794,878,820,887,895,945,841,900,907,953,917,959,968,988,853,914,922,964,932,969,971,997,943,977,981,1000,983,1004,1006,1016,868,927,940,973,949,978,982,1003,956,987,990,1007,993,1008,1011,1018,961,991,995,1010,998,1012,1013,1019,1001,1014,1015,1020,1017,1021,1022,1023}
以上结合图1至图5,对本申请实施例提供的极化码编码的方法作了详细说明。下面结合图6至图15对本申请实施例提供的极化码编码和译码的装置和设备进行说明。
图6为本申请实施例提供的极化码编码的装置300的示意性框图。如图6所示,装置300主要包括接收单元310和处理单元320。
获取单元310,用于获取所需母码长度对应的排序序列和待编码比特;
编码单元320,用于采用该所需母码长度对应的排序序列对待编码比特进行极化编码,得到编码后的序列。
其中,装置300可以是与终端通信的网络设备,也可以是一个终端。
参见图7,图7为本申请实施例提供的一种通信设备500,用于实现编码的功能。该通信设备包括处理装置504和收发器508。其中,处理装置504用于:
获取所需母码长度对应的排序序列和待编码比特;
采用该所需母码长度对应的排序序列对待编码比特进行极化编码,得到编码后的序列。
收发器508,用于发送该编码后的序列。
可选的,处理装置504可以是芯片或者集成电路。
通信设备500可以是与终端通信的网络设备,也可以是一个终端。
本申请实施例还提供一种用于编码的处理装置604,用于实现上述实施例中的极化码编码的方法。上述实施例的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。
可选地,当上述实施例中的极化码编码的方法全部或部分通过通过硬件实现时,参见图8所示,图8为处理装置504的内部结构示意图。处理装置504包括:
输入接口电路6142,用于获取所需母码长度对应的排序序列和待编码比特;
逻辑电路6144,用于采用所需母码长度对应的排序序列对待编码比特进行极化编码,得到编码后的序列。
输出接口电路6146,用于输出该编码后的序列。
上述逻辑电路6142可以用于执行图4中由发送设备执行编码方法。具体请见前面的方法实施例中描述,此处不再赘述。在具体实现时,上述处理装置可以是芯片或者集成电路。
可选地,当上述实施例的极化码编码的方法全部或部分通过软件实现时,处理装置604包括处理器5042和存储器5044。具体参见图9所示,图9为处理装置504的内部结构示意图。其中,
存储器5044,用于存储程序;
处理器5042,用于执行存储器中存储的程序,当程序被执行时,处理器5042用于执行上述实施例的极化码编码的方法。
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,具体参见图10所示,图10为处理装置的另一种内部结构示意图。
在另一种可选的实施例中,处理装置504也可以只包括处理器5042,上述存储器5042位于处理装置504之外,处理器5042通过电路/电线与存储器5044连接,用于读取并执行所述存储器5044中存储的程序。
图11为本申请实施例提供的极化码译码的装置600的示意性框图。如图11所示,装置600主要包括接收单元610和处理单元620。
获取单元610,用于获取待译码序列和所需母码长度对应的排序序列;
译码单元620,用于采用该所需母码长度对应的排序序列对待译码序列进行译码,得到译码后的序列。
参见图12,图12为本申请实施例提供的一种通信设备700,用于实现编码的功能。该通信设备包括处理装置704和收发器708。其中,
收发器708,用于接收待译码序列;
处理装置704,用于获取所需母码长度对应的排序序列,并采用该所需母码长度对应的排序序列对待译码序列进行译码,得到译码后的序列。
本申请实施例还提供了一种用于译码的处理装置704,用于实现上述实施例中的译码方法,上述是实施例的译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。该处理装置704的结构与前面编码装置中处理装置的结构相同,只是在实现的功能上有所不同,因此,此处只是对其区别进行说明。
参见图13,图13为处理装置704的一种内部结构示意图。
当处理装置704通过硬件实现时,该处理装置704中的逻辑电路7142用于采用所需母码长度对应的排序序列对输入的待译码序列进行译码,得到译码后的序列。
该逻辑电路7142可以用于执行图4由接收设备执行的译码方法,具体请见前面方法实施例中对译码侧的描述,此处不再赘述。在具体实现时,上述处理装置704可以是芯片或者集成电路。
上述是实施例的译码方法中的部分或全部通过软件来实现时,参见图14所示,图14为处理装置的另一种内部结构图。该处理装置中的处理器7042,用于执行存储器7044存储的程序,当所述程序被执行时,处理装置7042执行上述实施例中的译码方法。上述存储器7044可以是物理上独立的单元,也可以与处理器7042集成在一起。
在另一种可选的实施例中,处理装置也可以只包括处理器,上述存储器位于处理装置之外,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。
本申请实施例中用于编码的处理装置和用于译码的处理装置,在实际应用中可能是分别独立的。也有可能是集成在一起的,即形成一套装置。
上述通信设备可以是终端,也可以是网络设备。当该通信设备是终端之时,参见图15所示,图15为终端设备800的结构示意图。该终端800还可以包括电源812、用于给终端中的各种器件或电路提供电源。该终端还可以可以包括天线810,用于将收发器输出的上行数据通过无线信号发送出去,或者将收到的无线信号输出给收发器。
除此之外,为了使得终端的功能更加完善,该终端还可以包括输入单元814,显示单元816,音频电路818,摄像头820和传感器822等中的一个或多个,所述音频电路可以包括扬声器8182,麦克风8184等。
结合前面的描述,本领域的技术人员可以意识到,本文实施例的方法,可以通过硬件(例如,逻辑电路),或者软件,或者硬件与软件的结合来实现。这些方法究竟以硬件还是软件方式来执行,取决于技术方案的特定应用和设计约束条件。专业技术人员可以对每个特定的应用来使用不同方法来实现所描述的功能,但是这种实现不应认为超出本申请的范围。
当上述功能通过软件的形式实现并作为独立的产品销售或使用时,可以存储在一个计算机可读取存储介质中。在这种情况下,本申请的技术方案本质上或者说对现有技术做出贡献的部分或者该技术方案的部分可以以软件产品的形式体现出来,该计算机软件产品存储在一个存储介质中,包括若干指令用以使得一台计算机设备(可以是个人计算机,服务器,或者网络设备等)执行本申请各个实施例所述方法的全部或部分步骤。而前述的存储介质包括:U盘、移动硬盘、只读存储器(Read-Only Memory,ROM)、随机存取存储器(RandomAccess Memory,RAM)、磁碟或者光盘等各种可以存储程序代码的介质。
以上所述,仅为本申请的具体实施方式,但本申请的保护范围并不局限于此,任何熟悉本技术领域的技术人员在本申请揭露的技术范围内,可轻易想到变化或替换,都应涵盖在本申请的保护范围之内。因此,本申请的保护范围应以所述权利要求的保护范围为准。
Claims (14)
1.一种极化码编码的方法,其特征在于,所述方法包括:
发送设备获取所需母码长度对应的排序序列;
所述发送设备获取待编码比特;
所述发送设备采用所述所需母码长度对应的排序序列对所述待编码比特进行极化编码,得到编码后的序列。
2.根据权利要求1所述的方法,其特征在于,所述发送设备获取所需母码长度对应的排序序列,包括:
所述发送设备从长度为N0的序列中读取所述所需母码长度对应的序列,所述长度为N0的序列是最大母码长度对应的排序序列或可靠度归一化序列,其中,
所述最大母码长度对应的排序序列包括N0个极化信道的索引,所述N0个极化信道的索引是按照所述N0个极化信道的可靠度的大小排序的,
所述可靠度归一化序列包括N0个极化信道的索引,每个索引对应一个极化信道的可靠度的量化值,N0大于0且为整数。
3.根据权利要求2所述的方法,其特征在于,所述长度为N0的序列如序列1至序列32中的任意一个。
4.根据权利要求2或3所述的方法,其特征在于,所述最大母码长度对应的排序序列是通过将最大母码长度对应的初始排序序列中小于N的索引依次替换为第一序列包括的索引得到的,
其中,所述最大母码长度对应的初始排序序列包括N0个极化信道的索引,所述N0个极化信道的索引是按照所述N0个极化信道的可靠度的大小排序的,所述第一序列包括N个极化信道的索引,所述N个极化信道的索引是按照所述N个极化信道的可靠度的大小排序的,其中,N为正整数,且N≤N0。
5.根据权利要求4所述的方法,其特征在于,所述第一序列可以通过如下方式生成:
获取所述N个极化信道的极化权重,并将所述N个极化信道的索引按照所述N个极化信道的极化权重的大小进行排序,得到所述第一序列;或者,
根据至少一个第一指示序列对第一基础序列进行扩展,得到所述第一序列,其中,每个第一指示序列用于将一个长度等于n的序列扩展为长度等于2n的序列,所述第一基础序列包括R个极化信道的索引,所述R个极化信道的索引是按照所述R个极化信道的可靠度大小排序的,其中,n、R均为正整数,R<N0。
6.根据权利要求5所述的方法,其特征在于,所述第一基础序列可以通过如下方式生成:
获取所述R个极化信道的极化权重,并将所述R个极化信道的索引按照所述R个极化信道的极化权重的大小进行排序,得到所述第一基础序列。
7.根据权利要求2或3所述的方法,其特征在于,所述最大母码长度对应的初始排序序列可以通过如下方式生成:
根据至少一个第二指示序列,对第二基础序列进行扩展得到所述最大母码长度对应的初始排序序列,其中,所述第二基础序列包括T个极化信道的索引,所述T个极化信道的索引是按照所述T个极化信道的可靠度的大小排序的,每个第二指示序列用于将一个长度等于n的序列扩展为长度等于2n的序列,其中,所述n和T均正整数,T<N0。
8.一种编码的装置,其特征在于,包括:
获取单元,用于获取所需母码长度对应的排序序列和待编码比特;
编码单元,用于采用所述所需母码长度对应的排序序列对所述待编码比特进行极化编码,得到编码后的序列。
9.根据权利要求8所述的装置,其特征在于,所述获取单元具体用于从长度为N0的序列中读取所述所需母码长度对应的排序序列,所述长度为N0的序列是最大母码长度对应的排序序列或可靠度归一化序列,其中,
所述最大母码长度对应的排序序列包括N0个极化信道的索引,所述N0个极化信道的索引是按照所述N0个极化信道的可靠度的大小排序的,
所述可靠度归一化序列包括N0个极化信道的量化值,每个索引对应一个极化信道的可靠度的量化值,N0大于0且为整数。
10.根据权利要求9所述的装置,其特征在于,所述长度为N0的序列如序列1至序列32中的任意一个。
11.根据权利要求9或10所述的装置,其特征在于,所述最大母码长度对应的排序序列是通过将最大母码长度对应的初始排序序列中小于N的索引依次替换为第一序列包括的索引得到的,
其中,所述最大母码长度对应的初始排序序列包括N0个极化信道的索引,所述N0个极化信道的索引是按照所述N0个极化信道的可靠度的大小排序的,所述第一序列包括N个极化信道的索引,所述N个极化信道的索引是按照所述N个极化信道的可靠度的大小排序的,其中,N为正整数,且N≤N0。
12.根据权利要求11所述的装置,其特征在于,所述第一序列可以通过如下方式生成:
获取所述N个极化信道的极化权重,并将所述N个极化信道的索引按照所述N个极化信道的极化权重的大小进行排序,得到所述第一序列;或者,
根据至少一个第一指示序列对第一基础序列进行扩展,得到所述第一序列,其中,每个第一指示序列用于将一个长度等于n的序列扩展为长度等于2n的序列,所述第一基础序列包括R个极化信道的索引,所述R个极化信道的索引是按照所述R个极化信道的可靠度大小排序的,其中,n、R均为正整数,R<N0。
13.根据权利要求12所述的装置,其特征在于,所述第一基础序列可以通过如下方式生成:
获取所述R个极化信道的极化权重,并将所述R个极化信道的索引按照所述R个极化信道的极化权重的大小进行排序,得到所述第一基础序列。
14.根据权利要求9或10所述的装置,其特征在于,所述最大母码长度对应的初始排序序列可以通过如下方式生成:
根据至少一个第二指示序列,对第二基础序列进行扩展得到所述最大母码长度对应的初始排序序列,其中,所述第二基础序列包括T个极化信道的索引,所述T个极化信道的索引是按照所述T个极化信道的可靠度的大小排序的,每个第二指示序列用于将一个长度等于n的序列扩展为长度等于2n的序列,其中,所述n和T均正整数,T<N0。
Priority Applications (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
CN201710602871.4A CN109286403B (zh) | 2017-07-21 | 2017-07-21 | 极化码编码的方法和装置 |
Applications Claiming Priority (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
CN201710602871.4A CN109286403B (zh) | 2017-07-21 | 2017-07-21 | 极化码编码的方法和装置 |
Publications (2)
Publication Number | Publication Date |
---|---|
CN109286403A true CN109286403A (zh) | 2019-01-29 |
CN109286403B CN109286403B (zh) | 2022-05-10 |
Family
ID=65185391
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
CN201710602871.4A Active CN109286403B (zh) | 2017-07-21 | 2017-07-21 | 极化码编码的方法和装置 |
Country Status (1)
Country | Link |
---|---|
CN (1) | CN109286403B (zh) |
Cited By (4)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US10659194B2 (en) | 2017-08-02 | 2020-05-19 | Huawei Technologies Co., Ltd. | Polar code encoding method and apparatus in wireless communications |
WO2020253787A1 (zh) * | 2019-06-21 | 2020-12-24 | 中兴通讯股份有限公司 | 基站数据调制方法及装置 |
CN113078910A (zh) * | 2020-01-06 | 2021-07-06 | 大唐移动通信设备有限公司 | 一种比特位字段的确定方法、装置、介质和电子设备 |
US11811528B2 (en) | 2017-08-02 | 2023-11-07 | Huawei Technologies Co., Ltd. | Polar code encoding method and apparatus in wireless communications |
Citations (7)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20140208183A1 (en) * | 2013-01-23 | 2014-07-24 | Samsung Electronics Co., Ltd. | Method and system for encoding and decoding data using concatenated polar codes |
CN104539393A (zh) * | 2015-01-07 | 2015-04-22 | 北京邮电大学 | 一种基于极化码的信源编码方法 |
CN105811998A (zh) * | 2016-03-04 | 2016-07-27 | 深圳大学 | 一种基于密度演进的极化码构造方法及极化码编译码系统 |
CN106788456A (zh) * | 2016-12-14 | 2017-05-31 | 电子科技大学 | 一种极化码编译码方法 |
CN106789968A (zh) * | 2016-12-06 | 2017-05-31 | 哈尔滨工程大学 | 一种弱化窃听信道下基于极化码的安全编码方法 |
CN106899311A (zh) * | 2012-09-24 | 2017-06-27 | 华为技术有限公司 | 混合极性码的生成方法和生成装置 |
CN106953647A (zh) * | 2017-03-15 | 2017-07-14 | 中山大学 | 一种针对代数几何码的自适应Chase译码方法 |
-
2017
- 2017-07-21 CN CN201710602871.4A patent/CN109286403B/zh active Active
Patent Citations (7)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
CN106899311A (zh) * | 2012-09-24 | 2017-06-27 | 华为技术有限公司 | 混合极性码的生成方法和生成装置 |
US20140208183A1 (en) * | 2013-01-23 | 2014-07-24 | Samsung Electronics Co., Ltd. | Method and system for encoding and decoding data using concatenated polar codes |
CN104539393A (zh) * | 2015-01-07 | 2015-04-22 | 北京邮电大学 | 一种基于极化码的信源编码方法 |
CN105811998A (zh) * | 2016-03-04 | 2016-07-27 | 深圳大学 | 一种基于密度演进的极化码构造方法及极化码编译码系统 |
CN106789968A (zh) * | 2016-12-06 | 2017-05-31 | 哈尔滨工程大学 | 一种弱化窃听信道下基于极化码的安全编码方法 |
CN106788456A (zh) * | 2016-12-14 | 2017-05-31 | 电子科技大学 | 一种极化码编译码方法 |
CN106953647A (zh) * | 2017-03-15 | 2017-07-14 | 中山大学 | 一种针对代数几何码的自适应Chase译码方法 |
Non-Patent Citations (1)
Title |
---|
DAOLONG WU 等: "Ordered Statistic Decoding for Short Polar Codes", 《IEEE COMMUNICATIONS LETTERS》 * |
Cited By (6)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US10659194B2 (en) | 2017-08-02 | 2020-05-19 | Huawei Technologies Co., Ltd. | Polar code encoding method and apparatus in wireless communications |
US11165535B2 (en) | 2017-08-02 | 2021-11-02 | Huawei Technologies Co., Ltd. | Polar code encoding method and apparatus in wireless communications |
US11811528B2 (en) | 2017-08-02 | 2023-11-07 | Huawei Technologies Co., Ltd. | Polar code encoding method and apparatus in wireless communications |
WO2020253787A1 (zh) * | 2019-06-21 | 2020-12-24 | 中兴通讯股份有限公司 | 基站数据调制方法及装置 |
CN113078910A (zh) * | 2020-01-06 | 2021-07-06 | 大唐移动通信设备有限公司 | 一种比特位字段的确定方法、装置、介质和电子设备 |
CN113078910B (zh) * | 2020-01-06 | 2023-07-28 | 大唐移动通信设备有限公司 | 一种比特位字段的确定方法、装置、介质和电子设备 |
Also Published As
Publication number | Publication date |
---|---|
CN109286403B (zh) | 2022-05-10 |
Similar Documents
Publication | Publication Date | Title |
---|---|---|
CN108683479B (zh) | 一种速率匹配的方法和装置 | |
CN109889304A (zh) | 速率匹配方法、编码装置和通信装置 | |
CN109286403A (zh) | 极化码编码的方法和装置 | |
CN107124188A (zh) | 极化码的编码方法、译码方法、编码设备和译码设备 | |
CN108289010B (zh) | 一种数据处理方法和装置 | |
CN108234081B (zh) | 编码方法及装置 | |
CN109547034B (zh) | 译码方法及设备、译码器 | |
CN108809333B (zh) | 极化码编译码的方法、发送设备和接收设备 | |
CN111385059B (zh) | 极化编码调制的方法和装置 | |
CN109257140A (zh) | 一种极化信道可靠度排序的方法、极化码编码方法及装置 | |
CN108809482B (zh) | Polar码的速率匹配方法及装置 | |
CN107342846A (zh) | 一种编码方法、无线设备和芯片 | |
CN107800510A (zh) | 极化Polar码编码的方法及装置 | |
WO2020147526A1 (zh) | 一种级联crc码的极化码编码方法及装置 | |
CN109525360B (zh) | 极化码的速率匹配的方法和装置 | |
CN108574558A (zh) | 传输控制信息的方法和装置 | |
CN112600569A (zh) | 一种编码方法 | |
CN108809500A (zh) | 编码方法、装置和设备 | |
CN109150384A (zh) | 极化码编码的方法和装置 | |
CN110391874B (zh) | 极化码的速率匹配、解速率匹配方法及设备 | |
CN107437976A (zh) | 一种数据处理方法及设备 | |
CN108880743A (zh) | 一种Polar码传输方法及装置 | |
KR20190013374A (ko) | 통신 또는 방송 시스템에서 극부호 부호화/복호화 방법 및 장치 | |
CN108880565B (zh) | 极化码的编译码方法和通信设备 | |
CN109391358A (zh) | 极化码编码的方法和装置 |
Legal Events
Date | Code | Title | Description |
---|---|---|---|
PB01 | Publication | ||
PB01 | Publication | ||
SE01 | Entry into force of request for substantive examination | ||
SE01 | Entry into force of request for substantive examination | ||
GR01 | Patent grant | ||
GR01 | Patent grant |