CN108390740B - 一种信息的传输方法、译码方法和装置 - Google Patents

一种信息的传输方法、译码方法和装置 Download PDF

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CN108390740B
CN108390740B CN201710063758.3A CN201710063758A CN108390740B CN 108390740 B CN108390740 B CN 108390740B CN 201710063758 A CN201710063758 A CN 201710063758A CN 108390740 B CN108390740 B CN 108390740B
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张华滋
李榕
乔云飞
罗禾佳
张公正
陈莹
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Huawei Technologies Co Ltd
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Abstract

本申请提供一种信息的传输方法、译码方法和装置，该方法包括：发送设备根据预设的参数对的待编码的序列进行编码，得到编码后的序列，参数包括校验比特的数量、校验比特的位置以及校验方程，发送设备的参数需要预先进行设置，然后对待编码的序列先按照校验比特进行预编码，再进行polar编码，将编码后的序列发送至接收设备，有效降低数据传输的错误率，提高数据传输的可靠性。

Description

一种信息的传输方法、译码方法和装置
技术领域
本申请涉及通信技术，尤其涉及一种信息的传输方法、译码方法和装置。
背景技术
极化码(Polar Codes)是2008年由E.Ar1kan提出的一种新型信道编码。极化码基于信道极化(Channel Polarization)进行设计，是第一种能够通过严格的数学方法证明达到信道容量的构造性编码方案。Polar码是一种线性块码。其生成矩阵为GN,其编码过程为
Figure BDA0001220174590000011
其中
Figure BDA0001220174590000012
是一个二进制的行矢量，长度为N(即码长)；GN是一个N×N的矩阵，且
Figure BDA0001220174590000013
这里
Figure BDA0001220174590000014
定义为log2N个矩阵F2的克罗内克(Kronecker)乘积.
Figure BDA0001220174590000015
Figure BDA0001220174590000016
Figure BDA0001220174590000017
Polar码的编码过程中，
Figure BDA0001220174590000018
中的一部分比特用来携带信息，称为信息比特，这些比特的索引的集合记作A；另外的一部分比特置为收发端预先约定的固定值(称之为固定比特)，其索引的集合用A的补集Ac表示。不失一般性，这些固定比特通常被设为0，只需要收发端预先约定，固定比特序列可以被任意设置。从而，Polar码的编码输出可简化为：
Figure BDA0001220174590000021
这里uA为
Figure BDA0001220174590000022
中的信息比特集合，uA为长度K的行矢量，即|A|＝K，|·|表示集合中元素的个数，K为信息块大小，GN(A)是矩阵GN中由集合A中的索引对应的那些行得到的子矩阵，GN(A)是一个K×N的矩阵。Polar码的构造过程即集合A的选取过程，决定了Polar码的性能。
在提出Polar码之后，常用的信息传输方式是将待发送的信息块进行Polar编码得到编码块，然后进行传输，然而这种方式下的数据传输的错误率比较高，传输可靠性较低。
发明内容
本申请提供一种信息传输的方法和装置，用于解决常用的信息传输方式是将待发送的数据，即信息块进行Polar编码得到编码块，然后进行传输，然而这种方式下的数据传输的错误率比较高，传输可靠性较低的问题。
本申请第一方面提供一种信息传输的方法，包括：
发送设备根据预设的参数对的待编码的序列进行编码，得到编码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程；所述待编码的序列长度为K，所述K为大于0的整数；
所述发送设备将所述编码后的序列发送至接收设备。
在上述方案的具体实现中：
所述校验比特的数量是预先配置的，或者，所述校验比特的数量是根据译码器的列表长度确定的，或者，所述校验比特的数量是根据码长和码率确定的，或者，所述校验比特的数量是根据预先配置的比特数、译码器的列表长度、码长和码率确定的。
在上述方案中，校验比特为循环冗余校验(Cyclic Redundancy Check，CRC)比特时：
该校验比特的数量(也称为长度)可以被协议规定或者配置为2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,36,40,44,48,52,56,60,64等。校验比特的数量还可以根据list的大小确定，采用公式F1＝ceil(log2(L))，其中，L表示list的大小，F1表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。可能的list的大小(也称为长度)L为1，2，4，8，16，32，64，128，256，512，1024，与该些list的大小L对应的校验比特的数量F1为0，1，2，3，4，5，6，7，8，9，10。该校验比特的数量还可以通过码长和码率确定，其中“F2＝ceil(log2(M))”；或，“F2＝ceil(log2(K))”；或“F2＝ceil(a*log2(N))”，其中，a为系数，K表示带编码的序列长度(也称为信息比特长度)，M表示码长(编码后、速率匹配后)，F2表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。另外也校验比特的数量还可以是上述几种方式得到的数量的组合。
校验比特为奇偶校验(Parity Check，PC)比特时：
该校验比特的数量可以被配置为：2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,36,40,44,48,52,56,60,64等。校验比特的数量还可以根据list的大小确定，采用公式F1＝ceil(log2(L))，其中，L表示list的大小，F1表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。可能的list的大小(也称为长度)L为1，2，4，8，16，32，64，128，256，512，1024，与该些list的大小L对应的校验比特的数量F1为0，1，2，3，4，5，6，7，8，9，10。该校验比特的数量还可以通过码长和码率确定，PC数量＝Fp＝F2”，其中“F2＝ceil(log2(N)*(α-|α*(K/M-1/2)|2))”或“F2＝ceil(log2(M))”其中a为大于0的系数，通常可取0.75,1,1.25,1.5等，N表示母码长度(编码后、速率匹配前)，K表示带编码的序列长度(也称为信息比特长度)，M表示码长(编码后、速率匹配后)，F2表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。另外也校验比特的数量还可以是上述几种方式得到的数量的组合。
在一种具体实现方式中，当所述校验比特为CRC比特时，所述CRC比特的位置为：从一个新的生成矩阵中选择出的比特位置；所述新的生成矩阵是通过对CRC的生成矩阵进行行列转换，将其中只依赖部分信息比特的CRC比特置换至序号更小的子信道得到的。
在另一种实施方式中，当所述校验比特为PC比特时，所述PC比特的位置为：
预先设置的；所述PC比特的位置对应待编码序列中任意一个比特的位置，或者对应待编码序列多个比特的位置，所述多个比特中至少有两个比特的位置是非连续的；
或者，
根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
在上述任一方案的基础上，当所述校验比特为PC比特时，所述校验方程是根据校验方程表格确定的；其中，所述校验方程表格至少包括PC校验比特、信息比特以及参与校验方程的信息比特。在一种具体实现中，校验方程表格还包括固定比特、缩短比特。
本申请第二方面提供一种信息的传输方法，包括：
接收设备接收发送设备发送的编码后的序列；所述编码后的序列是根据预设的参数对待编码的序列进行编码后得到的，所述待编码的序列长度为K，所述K为大于0的整数；
所述接收设备根据预设的参数对所述编码后的序列进行译码，得到译码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程。
在上述方案中：
所述校验比特的数量是预先配置的，或者，所述校验比特的数量是根据译码器的列表长度确定的，或者，所述校验比特的数量是根据码长和码率确定的，或者，所述校验比特的数量是根据预先配置的比特数、译码器的列表长度、码长和码率确定的。
在上述方案中，校验比特为CRC比特时：
该校验比特的数量(也称为长度)可以被协议规定或者配置为2,3,4,5,6,7,8,9,10,11, 12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,36,40,44,48,52,56,60,64等。校验比特的数量还可以根据list的大小确定，采用公式F1＝ceil(log2(L))，其中，L表示list的大小，F1表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。可能的list的大小(也称为长度)L为1，2，4，8，16，32，64，128，256，512，1024，与该些list的大小L对应的校验比特的数量F1为0，1，2，3，4，5，6，7，8，9，10。该校验比特的数量还可以通过码长和码率确定，其中“F2＝ceil(log2(M))”；或，“F2＝ceil(log2(K))”；或“F2＝ceil(a*log2(N))”，其中，a为系数，K表示带编码的序列长度(也称为信息比特长度)，M表示码长(编码后、速率匹配后)，F2表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。另外也校验比特的数量还可以是上述几种方式得到的数量的组合。
校验比特为PC比特时：
该校验比特的数量可以被配置为：2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,36,40,44,48,52,56,60,64等。校验比特的数量还可以根据list的大小确定，采用公式F1＝ceil(log2(L))，其中，L表示list的大小，F1表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。可能的list的大小(也称为长度)L为1，2，4，8，16，32，64，128，256，512，1024，与该些list的大小L对应的校验比特的数量F1为0，1，2，3，4，5，6，7，8，9，10。该校验比特的数量还可以通过码长和码率确定，PC数量＝Fp＝F2”，其中“F2＝ceil(log2(N)*(α-|α*(K/M-1/2)|2))”或“F2＝ceil(log2(M))”其中a为大于0的系数，通常可取0.75,1,1.25,1.5等，N表示母码长度(编码后、速率匹配前)，K表示带编码的序列长度(也称为信息比特长度)，M表示码长(编码后、速率匹配后)，F2表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。另外也校验比特的数量还可以是上述几种方式得到的数量的组合。
在一种具体实施方式中，当所述校验比特为CRC比特时，所述CRC比特的位置为：
从一个新的生成矩阵中选择出的比特位置；所述新的生成矩阵是通过对CRC的生成矩阵进行行列转换，将其中只依赖部分信息比特的CRC比特置换至序号更小的子信道得到的。
在另一种实施方式中，当所述校验比特为PC比特时，所述PC比特的位置为：
预先设置的；所述PC比特的位置对应待编码序列中任意一个比特的位置，或者对应待编码序列多个比特的位置，所述多个比特中至少有两个比特的位置是非连续的；
或者，
根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
在上述任一方案的基础上，当所述校验比特为奇偶校验PC比特时，所述校验方程是 根据校验方程表格确定的；其中，所述校验方程表格至少包括PC校验比特、信息比特以及参与校验方程的信息比特。一种具体实现中，该校验方程表格还包括固定比特、缩短比特。
本申请第三方面提供一种信息的传输装置，包括：
处理模块，用于根据预设的参数对的待编码的序列进行编码，得到编码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程；所述待编码的序列长度为K，所述K为大于0的整数；
发送模块，用于将所述编码后的序列发送至接收设备。
在另一种实现方式中，当所述校验比特为CRC比特时，所述CRC比特的位置为：
从一个新的生成矩阵中选择出的比特位置；所述新的生成矩阵是通过对CRC的生成矩阵进行行列转换，将其中只依赖部分信息比特的CRC比特置换至序号更小的子信道得到的。
在另一种实现方式中，当所述校验比特为PC比特时，所述PC比特的位置为：
预先设置的；所述PC比特的位置对应待编码序列中任意一个比特的位置，或者对应待编码序列多个比特的位置，所述多个比特中至少有两个比特的位置是非连续的；
或者，
根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
在上述任一实施方式的基础上，当所述校验比特为PC比特时，所述校验方程是根据校验方程表格确定的；其中，所述校验方程表格可包括PC校验比特、信息比特、固定比特、缩短比特以及参与校验方程的信息比特。
本申请第四方面提供一种信息的传输装置，包括：
接收模块，用于接收发送设备发送的编码后的序列；所述编码后的序列是根据预设的参数对待编码的序列进行编码后得到的，所述待编码的序列长度为K，所述K为大于0的整数；
处理模块，用于根据预设的参数对所述编码后的序列进行译码，得到译码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程。
在一种实现方式中，当所述校验比特为CRC比特时，所述CRC比特的位置为：
从一个新的生成矩阵中选择出的比特位置；所述新的生成矩阵是通过对CRC的生成矩阵进行行列转换，将其中只依赖部分信息比特的CRC比特置换至序号更小的子信道得到的。
在另一种实现方式中，当所述校验比特为PC比特时，所述PC比特的位置为：
预先设置的；所述PC比特的位置对应待编码序列中任意一个比特的位置，或者对应待编码序列多个比特的位置，所述多个比特中至少有两个比特的位置是非连续的；
或者，
根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有信息子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
在上述任一方案的基础上，当所述校验比特为PC比特时，所述校验方程是根据校验方程表格确定的；其中，所述校验方程表格包括PC校验比特、信息比特、固定比特、缩短比特以及参与校验方程的信息比特。
本申请第五方面提供一种译码方法，接收设备从发送设备接收待译码的比特序列；所述待译码的比特序列中至少包括信息比特和至少一个校验比特；所述接收设备采用串行抵消列表SCL译码算法按顺序对所述待译码的比特进行译码，输出度量值最优的l条候选路径；所述1≤l≤L，所述L为SCL的宽度，所述L为大于或等于1的整数；
从度量值最优的候选路径开始，用所述至少一个校验比特对其中的T条候选路径进行校验，将校验通过的第一个候选路径中的比特序列作为译码输出；所述T为整数，所述1≤T≤l。
其中，度量值用于表示译码端计算得到的译码成功的概率，即“从度量值最优的候选路径开始”是指译码端计算得到的译码成功的概率最高的路径开始。
本申请第六方面提供一种译码装置，包括：
接收模块，用于从发送设备接收待译码的比特序列；所述待译码的比特序列中至少包括信息比特和至少一个校验比特；
处理模块，用于采用串行抵消列表SCL译码算法按顺序对所述待译码的比特进行译码，输出度量值最优的l条候选路径；所述1≤l≤L，所述L为SCL的宽度，所述L为大于或等于1的整数；
所述处理模块还用于从度量值最优的候选路径开始，用所述至少一个校验比特对其中的T条候选路径进行校验，将校验通过的第一个候选路径中的比特序列作为译码输出；所述T为整数，所述1≤T≤l。
本申请第七方面提供一种发送设备，包括：
处理器，用于根据预设的参数对的待编码的序列进行编码，得到编码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程；所述待编码的序列长度为K，所述K为大于0的整数；
发送器，用于将所述编码后的序列发送至接收设备。
在上述发送设备的具体实现方式中，还可以包括存储器，处理器的数量为至少一个，用来执行存储器存储的计算机执行指令。使得发送设备通过通信接口与接收设备之间进行数据交互来执行上述第一方面或者第一方面的各种实施方式提供的信息的传输方法。可选的，存储器可以集成在处理器的内部。
本申请第八方面提供一种接收设备，包括：
接收器，用于接收发送设备发送的编码后的序列；所述编码后的序列是根据预设的参数对待编码的序列进行编码后得到的，所述待编码的序列长度为K，所述K为大于0的整数；
处理器，用于根据预设的参数对所述编码后的序列进行译码，得到译码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程。
在上述接收设备的具体实现中，还可包括存储器，且处理器的数量为至少一个，用来执行存储器存储的执行指令。使得接收设备通过通信接口与发送设备之间进行数据交互来执行上述第二方面或者第二方面的各种实施方式提供的信息的传输方法，或者使得接收设备执行上述第五方面提供的译码方法。可选的，存储器还可以集成在处理器内部。
本申请第九方面提供一种可读存储介质，可读存储介质中存储有执行指令，当发送设备的至少一个处理器执行该执行指令时，发送设备执行上述第一方面或者第一方面的各种实施方式提供的信息的传输方法。
本申请第十方面提供一种可读存储介质，可读存储介质中存储有执行指令，当接收设备的至少一个处理器执行该执行指令时，接收设备执行上述第二方面或者第二方面的各种实施方式提供的信息的传输方法，或者接收设备执行上述第五方面或者第五方面的各种实施方式提供的译码方法。
本申请第九方面提供一种程序产品，该程序产品包括执行指令，该执行指令存储在可读存储介质中。发送设备的至少一个处理器可以从可读存储介质读取该执行指令，至少一个处理器执行该执行指令使得发送设备实施第一方面或者第一方面的各种实施方式提供的信息的传输方法。
本申请第十方面提供一种程序产品，该程序产品包括执行指令，该执行指令存储在可读存储介质中。接收设备的至少一个处理器可以从可读存储介质读取该执行指令，至少一个处理器执行该执行指令使得接收设备实施上述第二方面或者第二方面的各种实施方式提供的信息的传输方法，或者接收设备执行上述第五方面或者第五方面的各种实施方式提供的译码方法。
本申请提供的信息的传输方法、译码方法和装置，发送设备根据预设的参数对的待编码的序列进行编码，得到编码后的序列，参数包括校验比特的数量、校验比特的位置以及校验方程，发送设备的参数需要预先进行设置，然后对待编码的序列先按照校验比特进行预编码，再进行polar编码，将编码后的序列发送至接收设备，有效降低数据传输的错误率，提高数据传输的可靠性。
附图说明
图1为无线通信的基本流程示意图；
图2为采用CA-Polar的方式进行编码的示意图；
图3为采用PC-Polar(Parity-check-Polar)的方式进行编码的示意图；
图4为采用PC+CA-Polar的方式进行编码的示意图；
图5为本申请提供的信息的传输方法实施例一的流程示意图；
图6为校验方程的表格表示法的示意图；
图7为多项式[1,0,0,0,0]采用移位寄存器的示意图；
图8为多项式[1,0,1,0,0]采用移位寄存器的示意图；
图9为本申请的译码方法的译码过程示意图；
图10为本申请提供的信息的传输装置实施例的结构示意图；
图11为本申请提供的信息的传输装置实施例二的结构示意图；
图12为本申请提供的译码装置实施例的结构示意图。
具体实施方式
本申请实施例的技术方案可以应用5G通信系统或未来的通信系统，也可以用于其他各种无线通信系统，例如：全球移动通讯(Global System ofMobile communication，GSM)系统、码分多址(CDMA，Code Division Multiple Access)系统、宽带码分多址(Wideband Code Division Multiple Access，WCDMA)系统、通用分组无线业务(GeneralPacket Radio Service，GPRS)、长期演进(Long Term Evolution，LTE)系统、LTE频分双工(Frequency Division Duplex，FDD)系统、LTE时分双工(Time Division Duplex，TDD)、通用移动通信系统(Universal Mobile Telecommunication System，UMTS)等。
图1为无线通信的基本流程示意图，如图1所示，在发送端，信源依次经过信源编码、信道编码、数字调制后发出。在接收端，依次经过数字解调、信道译码、信源解码输出信宿。信道编码可以采用Polar码，而在信道译码的时候，可以采用SC译码、SCL译码等。为了提高Polar码的性能，现在又提出了很多在Polar码的基础上进行改进的技术，例如，CA-Polar码，PC-Polar码，CA+PC-Polar等等。
图2为采用CA-Polar的方式进行编码的示意图，即对待编码的信息块先进行级联循环冗余校验(Cyclic Redundancy Check，CRC)编码，然后再进行Polar码编码得到编码块进行发送，CA-Polar编码也称为CRC级联Polar编码。相应的，译码的时候可以采用CA-SCL(CRC-Aided Successive Cancellation List)译码算法进行译码，即在SCL译码输出的多条候选路径中选择CRC通过的候选路径作为译码输出。采用CA-SCL译码算法，能显著提高Polar码的性能。CA-Polar可以由以下参数唯一确定：CRC校验比特的数量、CRC校验比特的位置和CRC校验方程。
图3为采用PC-Polar(Parity-check-Polar)的方式进行编码的示意图。PC-Polar的主要思想是选择一些校验冻结比特，也称为奇偶校验比特(Parity-check-frozen)或动态冻结(Dynamic Frozen)比特，分布到信息块中，这些校验冻结比特的值由其前面的信息比特根据校验方程确定。PC-Polar的译码可以基于SCL译码算法。信息比特和静态冻结比特的处理与SCL译码算法的处理一样，不同是校验冻结比特的处理。具体来说是利用校验冻结比特在译码过程中完成排序、剪枝的过程，PC-SCL译码最终输出最可靠的路径。PC-Polar由以下参数唯一确定：校验冻结比特的数量，校验冻结比特的位置和校验方程。
图4为采用PC+CA-Polar的方式进行编码的示意图。编码过程包括：(1)对待编码的信息块先进行级联循环冗余校验(Cyclic Redundancy Check，CRC)编码；(2)进行PC编码，即确定信息比特、静态冻结比特和校验冻结比特的值；(3)然后进行Polar码编码得到编码块。CRC编码矩阵由如下参数唯一确定：CRC校验比特的数量、CRC校验比特的位置和CRC校验方程。而PC编码是由如下参数确定：校验冻结比特的数量，校 验冻结比特的位置和校验方程。
综上所述，对于校验级联Polar编码(CA-Polar，PC-Polar或PC+CA-Polar)，编码矩阵可由以下参数唯一决定，即：校验比特的数量、校验比特的位置，校验方程的选择。这些参数的确定有很多种方式，在下面的实施例中将会介绍，后续的技术方案中，应理解：若不区分CRC比特和PC比特的话，则CRC的也适用于PC，PC的也适用于CRC。
图5为本申请提供的信息的传输方法实施例一的流程示意图，如图4所示，在图1所示的通信流程示意图基础上，或者其他进行信息交互的发送设备和接收设备之间，本申请提供的信息的传输方法具体包括以下步骤：
S101：发送设备根据预设的参数对的待编码的序列进行编码，得到编码后的序列；参数包括校验比特的数量、校验比特的位置以及校验方程。
上述校验比特是用于对信息比特进行校验的。能够实现校验的方式可以有很多种，例如，采用CA-Polar编码方式，或者PC-Polar编码方式，或者PC+CA-Polar编码方式等，但是本领域的技术人员可以理解，在实现时可以不限于这些编码方式，只要在编码过程中采用了校验比特对信息比特进行了校验即可。
在本步骤中，发送设备需要对待发送的序列进行编码处理，为了能使接收设备正确解码发送设备发送的编码序列，发送设备和接收设备需要事先约定好编码过程中使用的参数，例如校验比特的数量、校验比特的位置以及校验方程；或者发送设备需要在接收设备译码之前，将这些参数通知给接收设备。上述参数可以是协议中规定好，并且配置在发送设备上的，也可以是发送设备与接收设备之间协商得到的，或者是其他设备发送的通知消息中携带的，对此本方案不做限制。
在该方案的具体实现中，校验比特的数量的确定方式至少包括以下几种：
该校验比特的数量是预先配置的；或者，校验比特的数量是根据译码器的列表长度确定的；或者，所述校验比特的数量是根据码长和码率确定的；或者，所述校验比特的数量是根据预先配置的比特数、译码器的列表长度、码长和码率确定的，即综合前三种方式得到的数量整体作为校验比特的数量(也称为校验比特的长度)；或者，所述校验比特的数量为J，所述J≥log2(T/FAR)；所述T为允许使用校验比特进行校验的最大次数，所述FAR为使用校验比特进行单次检验的最大虚警概率；或者，所述校验比特的数量为J(T)，所述J(T)＝J(1)+ΔJ＝J(1)+log2T；其中，所述J(1)为单次校验是需要使用的校验比特的数量，所述J(T)为T次校验需要使用的校验比特的数量，所述T为允许使用校验比特进行校验的最大次数，其中，T＝1时候，一次校验指的是该路径对应的译码结果的一次检错。
校验比特的位置可以是待编码的序列按照序号大小排序之后，连续选择的与校验比特的数量相同的位置，或者也可以是随机选择出位置，还可以是直接预先设置的配置，还可以是根据预设的位置参数确定的位置。
具体的，校验比特的位置的确定方式至少包括以下几种方式：
第一种方式，校验比特的位置为：在待编码序列中按照比特的位置序号从大到小的顺序，从位置序号为NX的比特开始选择的连续FP个比特的位置；或者在待编码序列中 按照比特的位置序号从小到大的顺序，从位置序号为NX的比特开始选择的连续FP个比特的位置；其中，0≤NX≤K，0<Fp<K，待编码序列中比特的位置序号为大于等于0且小于K的任意整数。
第二种方式，校验比特的位置为：从一个新的生成矩阵中选择出的比特位置；所述新的生成矩阵是通过对校验生成矩阵进行行列转换，将其中只依赖部分信息比特的校验比特置换至序号更小的子信道得到的位置。
第三种方式，校验比特的位置对应待编码序列中任意一个比特的位置。
第四种方式，校验比特的位置对应待编码序列多个比特的位置，所述多个比特中至少有两个比特的位置是非连续的。
第五种方式，校验比特的位置是根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度
同样的，校验方程可以预先进行配置，或者根据多项式或者校验方程表格进行确定，校验方程的多项式可以采用二进制，八进制，十进制，十六进制，位置向量等中的任一种形式进行表示。
发送设备在得到需要的校验比特的参数之后，对待编码的序列先通过校验矩阵进行编码，然后再进行Polar编码，得到编码后的序列。作为一种可选的实施例，校验矩阵也可以与Polar码的矩阵合成一个矩阵，即对待编码的采用合成后的矩阵进行编码后就得到了编码后的序列。
S102：发送设备将编码后的序列发送至接收设备。
S103：接收设备根据预设的参数或者接收到的参数对编码后的序列进行译码，得到译码后的序列；该预设的参数或者接收到的参数包括校验比特的数量、校验比特的位置以及校验方程。
在上述步骤中，发送设备将编码后的序列发送给接收设备，即接收设备接收到发送设备发送的编码后的序列，接收设备需要进行译码处理得到译码后的序列，因此该接收设备也需要校验比特的数量，校验比特的位置以及校验方程来进行译码。
上述预设的参数可以是协议中规定好，并且配置在接收设备上的，也可以是发送设备与接收设备之间协商得到的，或者是发送设备或其他设备发送给接收设备的通知消息中携带的，对此本方案不做限制。
接收设备译码时采用的校验比特的数量、校验比特的位置和校验方程与发送设备相同，关于这些参数是如何确定的前面已经描述过，此处不再详述。
本实施例提供的信息的传输方法，发送设备根据预设的参数对的待编码的序列进行编码，得到编码后的序列，参数包括校验比特的数量、校验比特的位置以及校验方程，发送设备的参数需要预先进行设置，然后对待编码的序列先按照校验比特进行预编码，再进行polar编码，将编码后的序列发送至接收设备，有效降低信息传输的错误率，提高信息传输的可靠性。
结合上述实施例，下面具体介绍发送设备在信息传输时的编码过程中使用的参数是如 何确定的。以CRC校验或者PC校验为例，对校验比特的数量、校验比特的位置以及校验方程的确定方式进行具体说明。例如：与CRC相关的参数的确定方法可以适用于CA-Polar和CA+PC-Polar两种编码方式，与PC相关的参数的确定方法可以适用于PC-Polar和CA+PC-Polar两种编码方式。与CRC相关的参数包括：CRC比特的数量、CRC比特的位置，CRC校验方程。与PC相关的参数包括：PC比特的数量、PC比特的位置，PC校验方程。
一、校验比特的数量(也称为校验比特的长度)
对于在编码过程中采用CRC比特进行校验的编码方式，例如，CA-Polar或者CA+PC-Polar编码方式，CRC比特的数量可以采用如下几种方式中的任意一种来确定：
(1)、CRC比特的数量是预先配置的，即预先固定CRC长度。
预先在设备中配置CRC比特的长度(即CRC比特的数量)为固定值，或者在协议中规定CRC比特的长度(即CRC比特的数量)为固定值：例如“CRC比特数量J＝F0”，例如，，F0＝2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,36,40,44,48,52,56,60,64等。
(2)、CRC比特的数量可以根据译码器的列表(list)长度确定。
例如，CRC比特的数量J＝F1；
其中，“F1＝ceil(log2(L))”，L表示list的大小，F1表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。可能的list的大小(也称为长度)L为1，2，4，8，16，32，64，128，256，512，1024，与该些list的大小L对应的校验比特的数量F1为0，1，2，3，4，5，6，7，8，9，10。
或者，在上述方式(1)的F0的基础上，再增加一个F1作为CRC的长度，即CRC比特的数量J＝F0+F1。
(3)、CRC比特的数量可以是根据码长和码率确定。
例如：“CRC比特数量J＝F2”，其中“F2＝ceil(log2(M))”或“F2＝ceil(log2(K))”或“F2＝ceil(a*log2(N))”；其中，a为系数，K表示待编码的序列长度(也称为信息比特长度)，M表示码长(编码后、速率匹配后)，F2表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。一种具体实现中，F2可以是：3,4,5,6,7,8,9,10,11。
(4)、CRC比特的数量可以根据预先配置的比特数、译码器的列表长度、码长和码率来确定。
例如，校验比特的数量还可以是上述几种方式得到的数量的组合，即CRC比特的数量J＝F0+F1+F2。
对于在编码过程中采用PC比特进行校验的编码方式，例如，PC-Polar或者CA+PC-Polar编码方式，PC比特(也可以称为奇偶校验比特，或者奇偶校验冻结比特，或者动态冻结比特)的数量可以采用如下几种方式中的一种来确定：(1)、PC比特的数量可以是预先配 置的，即预先固定PC数量：
预先在设备中配置PC比特的长度(即PC比特的数量)为固定值，或者在协议中规定PC比特的长度(即PC比特的数量)为固定值：例如：“PC比特的数量P＝F0”，例如，F0＝2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,36,40,44,48,52,56,60,64等。
(2)、PC比特的数量可以是根据译码器的列表(list)长度确定。
例如，PC比特的数量P＝F1；
其中，“F1＝ceil(log2(L))”，L表示list的大小，F1表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。可能的list的大小(也称为长度)L为1，2，4，8，16，32，64，128，256，512，1024，与该些list的大小L对应的校验比特的数量F1为0，1，2，3，4，5，6，7，8，9，10。
或者，在上述方式(1)的F0的基础上，再增加一个F1作为PC的长度，即CRC比特的数量P＝F0+F1。
(3)、PC比特的数量可以是根据码长和码率确定的。
例如：PC比特的数量P＝F2，其中，F2＝ceil(log2(N)*(α-|α*(K/M-1/2)|2))或F2＝ceil(log2(M))；
其中a为大于0的系数，通常可取0.75,1,1.25,1.5等，N表示母码长度(编码后、速率匹配前)，K表示待编码的序列长度(也称为信息比特长度)，M表示码长(编码后、速率匹配后)，F2表示校验比特的数量，函数ceil表示向上取整，在实际应用中也可以是向下取整，还可以是四舍五入。
(4)、PC比特的数量可以是根据预先配置的比特数、译码器的列表长度、码长和码率确定的。
即PC比特的数量为上述参数的组合：如PC比特的数量P＝F0+F1+F2。
二、校验比特的位置
对于在编码过程中采用CRC比特进行校验的编码方式，例如，CA-Polar或者CA+PC-Polar编码方式，CRC比特的位置可以采用如下几种方式中的任意一种来确定：
(1)、CRC比特的位置是连续的
例如，待编码序列中比特的位置序号为大于等于0且小于待编码序列的长度K的任意整数，即带编码序列中的每个比特都有唯一的序号，该序号为连续、且大小不同的数字，那么一种CRC比特的位置的具体确定方式为：可以在待编码序列中按照比特的位置序号从大到小的顺序，从位置序号为NX的比特开始选择的连续J个比特的位置作为CRC比特的位置；或者，可以在待编码序列中按照比特的位置序号从小到大的顺序，从位置序号为NX的比特开始选择的连续J个比特的位置作为CRC比特的位置。其中，0≤NX＜K，0<J<K。例如，选择最靠后的J个连续位置作为CRC比特的位置。
(2)、CRC比特的位置是分散的
该方式中，CRC比特的位置可以为从一个新的生成矩阵中选择出的比特位置；所述新的生成矩阵是通过对CRC的生成矩阵进行行列转换，将其中只依赖部分信息比特的CRC比特置换至序号更小的子信道得到的。
例如：将CRC比特的生成矩阵做基本行列变换(如高斯消元)和/或行列置换(如列重排)，后使得一些CRC比特只依赖于部分信息比特或者只和部分信息比特相关，因此可以将此部分CRC比特置换到序号更小的子信道(例如，将此部分CRC比特在CRC生成矩阵中对应的列与所述部分信息比特后面一列进行置换)，在译码时帮助提前排除错误路径。
以3比特CRC举例，若其生成矩阵如下：
Figure BDA0001220174590000131
可以发现，3个CRC比特中的最后一个比特(对应生成矩阵第6列)只和前两个比特相关，因此可以将该最后一个CRC比特在CRC生成矩阵中对应的列提前至第三列：
Figure BDA0001220174590000132
这样，当接收端译到第三个比特时，可将其看做一个校验比特，并将其置为前两个比特的异或值，并剪去不满足预设值的路径。
上述实例中是以校验比特的数量是3比特为例进行说明的，具体的实现中，校验比特的数量本方案不做限制。
分散CRC比特的位置还可以通过其它方法来确定，例如，通过高斯消元简化CRC生成矩阵，然后通过行列置换将CRC比特中只依赖于部分信息比特的位置提前，
如xp＝(ui+uj+…+uk)，即CRC码字第p个比特为第i,j,…,k的信息比特的模2和，则将CRC比特中的第p个比特前移到第max{i,j,…,k}+1个位置，其中，xp表示第p个校验比特的校验位的值，ui表示第i个信息比特的译码值,i,j,……k表示信息比特的序号,。
对于在编码过程中采用PC比特进行校验的编码方式，例如，PC-Polar或者CA+PC-Polar编码方式，PC比特的位置可以采用如下几种方式中的任意一种来确定：
(1)、PC比特的位置是连续
该方式的含义是，待编码序列中比特的位置序号为大于等于0且小于待编码序列的长度K的任意整数，即带编码序列中的每个比特都有唯一的序号，该序号为连续、且大小不同的数字，那么一种PC比特的位置的具体确定方式为：可以在在待编码序列中按照比特的位置序号从大到小的顺序，从位置序号为NX的比特开始选择的连续P个比特的位置作为PC比特的位置；或者，可以在待编码序列中按照比特的位置序号从小到大的顺序，从位置序号为NX的比特开始选择的连续P个比特的位置作为PC比特的位置。其中，0≤NX≤K，0<P<K。例如，可以选择CRC比特之前的P个连续位置作为PC比特的位置。
(2)PC比特的位置是任意的
例如，可以预设任一的PC比特的位置，对于没有规律可循的PC位置，可以将所有PC位置存表，该方式通常在超短码中使用。一般来说，PC比特的位置对应待编码序列中任意一个比特的位置，或者对应待编码序列多个比特的位置，所述多个比特中至少有两个比特的位置是非连续的。
(3)PC比特的位置是根据子信道行重选择的
对于长码，存储所有PC比特的位置会带来较大存储开销，可通过一些规律来减少存储开销，以及加速硬件实现。该方案中，PC比特的位置可以根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有信息子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。具体的确定过程包括：
首先，定义归一化可靠度序列Zi：给定按可靠度从小到大已排序的子信道序号序列Q，以及序列长度Nmax，则归一化可靠度可唯一定义为：for ifrom 0to Nmax-1,set Z[Qi]＝i；endfor
将子信道对应行重RWi的定义为：Polar码生成矩阵中，该(第i个)子信道对应的行中，1的个数。
将子信道序号的汉明重量HWi的定义为：该(第i个)子信道的序号的2进制表示中，1的个数。例如，第5个子信道的序号用2进制可以表示为101，则HW5＝2。对于所有i满足RWi＝2^(HWi)。因HW的取值范围更小，因此在标准和硬件中更可能用HW而不是RW来表示。
对于一个PC-Polar码，其信息比特/冻结比特/校验比特位置可由2或3个参数唯一确定：
信息阈值ZI:表示所有信息子信道中(除去缩短位置)，可靠度最低子信道对应的可靠度(或极化权重)。
校验阈值ZP:表示所有校验子信道中(除去缩短位置)，可靠度大于最不可靠信息比特(>ZI)的，且行重在其中最大的子信道集合中，可靠度最低子信道对应的可靠度。
(可选的)校验行重RWP(或校验汉明重HWP)表示可靠度等于校验阈值ZP的子信道对应的行重(或校验汉明重)。其可通过ZP唯一换算得到，也可作为参数直接给出。
给定了(ZI,ZP,RWP)，以及第i个子信道的行重RWi以及行重Zi，即可获得潜在的PC比特可选位置(缩短位置不在可选范围之内)，且可以分为三类：
A类PC比特位置：子信道行重为RWP/2，且可靠度不低于信息阈值的位置(Z≥ZI&&RW＝RWP/2)，以及行重为RWP，且可靠度不低于校验阈值的位置(Z≥ZP&&RW＝RWP)。
B类PC比特位置：所有可靠度低于信息阈值的子信道位置(Z<ZI)。
W类PC比特位置：为以下两种位置的并集(一)所有A类子信道位置，(二)B类子信道位置中，行重为为RWP/2以及行重为RWP的位置。即((RW＝RWP/2)||(Z<ZI&&RW＝RWP)||(Z>ZP&&RW＝RWP))。
给定了上述PC比特的位置分类后，可以使用以下方式来确定PC比特的位置，以及PC-Polar码中其它比特的位置：
(1)A类校验:只选择A类PC比特位置(可靠度较高的位置)，包括：
F(冻结比特):Zi<ZI；
PF(PC比特):(RWi＝＝RWP/2&&Zi>ZI)||(RWi＝＝RWP&&Zi>ZP)；I(信息比特):Zi≥ZI&&I≠(PF||P)。
(2)B类校验：只选择B类PC比特位置(可靠度较低的位置)，包括：
PF(PC比特):Zi<ZI；
I(信息比特):Zi≥ZI&&I≠(PF||P)；
(3)W类校验:选择W类PC比特的位置；
F(冻结比特):Zi<ZI&&RWi≠RWP；
PF(PC比特):(RWi＝＝RWP/2)||(RWi＝＝RWP&&Zi>ZP)||(Zi<ZI&&RW＝＝RWP)；
I(信息比特):Zi≥ZI&&I≠(PF||P)。
(4)全校验:选择所有非信息比特且非缩短子信道的位置，包括：
PF(PC比特):Zi<ZI||(RWi<RWP)||(RWi＝＝RWP&&Zi>ZP)；或者，
I(信息比特):Zi≥ZI&&I≠(PF||P)；或者，
(5)混合校验:将CRC校验与上述任一中PC比特的位置产生的校验码级联。
下面将归一化可靠度序列和子信道行重序列可能的情况进行举例：
(一)、归一化可靠度序列Zi：
Zi(8)：0,1,2,4,3,5,6,7；
Zi(16)：0,1,2,5,3,6,7,11,4,8,9,12,10,13,14,15；
Zi(32)：0,1,2,6,3,7,8,15,4,9,10,17,12,18,20,26,5,11,13,19,14,21,22,27,16,23,24,28，25,29,30,31；
Zi(64)：0,1,2,6,3,8,9,18,4,10,11,21,13,23,25,37,5,12,14,24,16,27,29,41,19,30,32,43，35,46,48,56,7,15,17,28,20,31,33,44,22,34,36,47,39,49,51,58,26,38,40,50,42,52,53,59,45,54,55,60,57,61,62,63；
Zi(128)：0,1,2,6,3,8,9,19,4,10,12,23,14,26,29,46,5,13,15,27,17,31,34,52,20,35,38，56,42,61,65,84,7,16,18,33,21,36,39,58,25,41,45,64,49,68,72,90,30,47,51,70,54,74,77,95,60,79,83,99,87,103,105,116,11,22,24,40,28,44,48,67,32,50,53,73,57,76,80,97,37,55,59,78,63,82,86,102,69,88,91,106,94,109,111,120,43,62,66,85,71,89,92,107,75,93,96,110,100,112,114,122,81,98,101,113,104,115,117,123,108,118,119,124,121,125,126,127；
Zi(256)：0,1,2,6,3,8,9,20,4,10,12,24,14,27,30,51,5,13,15,28,18,33,37,59,21,38,42，65,47,72,78,108,7,17,19,36,22,40,43,68,26,46,50,77,55,83,88,120,32,53,58,86,63,92,97,130,71,100,107,138,114,146,152,182,11,23,25,44,29,49,54,81,34,56,60,89,66,94,101,133,41,64,69,98,76,106,112,144,84,115,121,153,127,160,165,194,48,74,80,110,87,118,124,156,93,126,132,164,139,170,176,203,104,136,142,173,150,180,185,210,159,188,193,216,198,220,224,239,16,31,35,57,39,62,67,96,45,70,75,105,82,113,119,151,52,79,85,116,91,123,129,162,99,131,137,168,145,175,181,207,61,90,95,128,102,134,140,171,111,143,149,179,157,186,191,214,122,154,161,189,166,195,199,221,174,201,206,226,211,230,232,244,73,1 03,109,141,117,148,155,184,125,158,163,192,169,197,202,223,135,167,172,200,178,205,209,229,187,212,215,233,219,236,238,248,147,177,183,208,190,213,217,234,196,218,222,237,227,240,242,250,204,225,228,241,231,243,245,251,235,246,247,252,249,253,254,255；
Zi(512)：0,1,2,6,3,8,9,20,4,10,12,24,14,28,31,54,5,13,15,29,18,34,38,63,21,39，43,70,49,78,85,124,7,17,19,37,22,41,44,73,27,48,53,84,59,92,98,141,33,56,62,95,68,103,110,155,77,114,123,168,133,180,190,242,11,23,26,46,30,52,58,90,35,60,65,100,71,107,116,161,42,69,75,112,83,122,131,178,93,134,143,192,152,203,213,267,51,81,88,128,97,139,147,197,106,151,159,211,170,222,232,287,120,165,175,227,188,240,249,303,202,254,265,317,277,329,339,386,16,32,36,61,40,67,72,109,47,76,82,121,91,132,140,189,55,86,94,135,102,145,154,205,113,157,166,217,179,230,241,295,66,101,108,153,117,162,171,223,130,177,187,239,199,251,261,313,144,193,204,256,214,268,278,330,228,282,293,344,304,355,363,407,80,119,127,174,138,186,196,248,150,201,210,264,221,276,286,338,164,215,226,280,238,292,302,353,253,305,316,365,328,375,385,424,184,236,245,299,259,311,320,369,274,326,335,382,348,393,400,437,291,342,351,396,362,406,412,447,374,415,422,454,432,461,466,486,25,45,50,79,57,89,96,137,64,99,105,149,115,160,169,220,74,111,118,163,129,176,185,237,142,191,200,252,212,266,275,327,87,126,136,183,146,195,206,258,158,209,219,273,231,285,296,347,173,225,235,290,247,301,310,361,263,315,325,373,337,384,392,431,104,148,156,207,167,218,229,283,181,233,243,297,255,307,318,367,198,250,260,312,272,324,334,381,288,340,349,394,358,403,410,445,216,270,281,332,294,345,354,398,306,357,366,409,376,417,425,456,322,371,379,420,390,429,435,464,402,439,444,471,450,475,479,495,125,172,182,234,194,246,257,309,208,262,271,323,284,336,346,391,224,279,289,341,300,352,360,405,314,364,372,414,383,423,430,460,244,298,308,359,319,368,377,418,333,380,389,428,399,436,442,469,350,395,404,440,411,446,451,476,421,453,459,481,465,485,488,500,269,321,331,378,343,388,397,434,356,401,408,443,416,449,455,478,370,413,419,452,427,458,463,484,438,467,470,489,474,492,494,504,387,426,433,462,441,468,472,490,448,473,477,493,482,496,498,506,457,480,483,497,487,499,501,507,491,502,503,508,505,509,510,511；
Zi(1024)：0,1,2,6,3,8,9,20,4,10,12,24,14,28,31,55,5,13,15,29,18,34,38,64,21,39,44，72,50,81,89,133,7,17,19,37,22,42,45,76,27,49,54,88,60,97,103,154,33,57,63,100,70,109,117,171,80,121,132,187,144,202,215,286,11,23,26,47,30,53,59,94,35,61,66,105,73,113,123,177,43,71,78,119,87,131,141,199,98,145,156,217,167,232,245,322,52,85,92,138,102,151,161,224,112,166,175,243,189,258,272,353,129,184,196,266,212,283,297,380,231,304,320,403,338,423,440,530,16,32,36,62,41,69,75,116,48,79,86,130,96,143,153,214,56,90,99,146,108,159,170,235,120,173,185,252,201,270,285,366,67,106,114,168,125,180,191,261,140,198,211,282,227,300,315,398,157,218,233,306,248,325,341,426,267,346,363,449,382,470,487,576,84,128,137,195,150,210,223,295,165,230,242,319,257,337,352,439,183,250,265,344,281,362,378,466,303,384,402,490,422,510,529,617,207,278,291,374,311,393,410,499, 334,419,435,525,455,545,561,650,360,446,463,552,483,572,589,675,508,596,614,697,635,716,733,804,25,46,51,83,58,93,101,149,65,104,111,164,122,176,188,256,77,118,127,182,139,197,209,280,155,216,229,302,244,321,336,421,91,136,147,206,160,222,236,310,174,241,254,333,271,351,367,454,194,264,277,359,294,377,392,482,318,401,418,507,438,528,544,634,110,162,172,238,186,253,268,347,203,273,287,369,305,386,404,492,226,299,314,397,332,417,433,523,355,442,458,548,476,566,582,669,251,329,345,430,364,450,468,556,385,474,491,580,511,599,618,700,413,503,519,609,539,629,644,727,564,653,667,749,684,764,777,845,135,193,205,276,221,293,309,391,240,317,331,416,350,437,453,543,263,343,358,445,376,465,481,571,400,489,506,595,527,616,633,715,289,372,387,477,408,497,514,603,432,522,538,628,559,648,662,744,461,551,567,655,587,674,688,768,612,696,711,786,731,803,815,875,328,412,429,518,448,537,554,642,473,563,579,666,598,683,699,776,502,592,608,693,627,710,725,798,652,735,748,819,763,833,844,899,535,624,640,722,658,739,754,823,681,761,774,842,789,854,865,916,707,784,795,860,810,871,881,928,831,889,897,941,908,949,955,983,40,68,74,115,82,126,134,192,95,142,152,213,163,228,239,316,107,158,169,234,181,249,262,342,200,269,284,365,301,383,399,488,124,179,190,260,204,275,288,371,225,298,313,396,330,415,431,521,247,324,340,425,357,444,460,550,381,469,486,575,505,594,611,695,148,208,220,292,237,312,327,411,255,335,349,436,368,456,472,562,279,361,375,464,395,485,501,591,420,509,526,615,546,636,651,734,308,390,407,496,428,517,534,623,452,542,558,647,578,665,680,760,480,570,586,673,607,692,706,783,632,714,730,802,747,818,830,888,178,246,259,339,274,356,370,459,296,379,394,484,414,504,520,610,323,405,424,512,443,532,549,638,467,555,573,659,593,678,694,772,354,441,457,547,475,565,581,668,500,590,606,691,626,709,724,797,531,619,637,718,654,736,750,820,676,755,770,837,785,851,861,913,389,479,495,585,516,605,622,705,541,631,646,729,664,746,759,829,569,656,672,752,690,769,782,849,713,787,801,863,817,876,887,932,602,687,702,779,721,794,807,868,743,814,826,884,841,896,905,946,767,835,847,901,859,912,919,958,874,922,930,965,940,972,977,998,219,290,307,388,326,409,427,515,348,434,451,540,471,560,577,663,373,462,478,568,498,588,604,689,524,613,630,712,649,732,745,816,406,494,513,601,533,621,639,720,557,645,661,742,679,758,773,840,584,671,686,766,704,781,793,858,728,800,813,873,828,886,895,939,447,536,553,641,574,660,677,756,597,682,698,775,717,790,805,866,625,708,723,796,741,812,825,883,762,832,843,898,855,909,917,956,657,738,753,822,771,838,850,903,788,853,864,915,877,924,933,967,809,870,880,927,893,937,944,975,907,948,954,982,961,987,991,1007,493,583,600,685,620,703,719,792,643,726,740,811,757,827,839,894,670,751,765,834,780,848,857,911,799,862,872,921,885,931,938,971,701,778,791,856,806,867,878,925,824,882,892,936,904,945,952,980,846,900,910,950,918,957,962,988,929,964,970,993,976,997,1000,1012,737,808,821,879,836,891,902,943,852,906,914,953,923,960,966,990,869,920,926,963,935,969,974,996,947,978,981,1001,986,1004,1006,1016,890,934,942,973,951,979,984,1002,959,985,989,1005,994,1008,1010,1018,968,992,995,1009,999,1011,1013,1019,1003,1014,1015,1020,1017,1021,1022,1023；
Zi(2048)：0,1,2,6,3,8,9,20,4,10,12,24,14,28,31,55,5,13,15,29,18,34,38,64,21,39，44,73,50,82,90,137,7,17,19,37,22,42,45,77,27,49,54,89,60,98,105,160,33,57,63,101,71,111,120,178,81,124,136,196,149,213,229,312,11,23,26,47,30,53,59,95,35,61,67,107,74,116,127,186,43,72,79,122,88,135,146,210,99,150,162,231,174,248,264,357,52,86,93,143,103,156,167,239,115,173,184,262,199,279,296,397,133,193,207,288,225,308,326,431,247,334,355,462,377,488,512,641,16,32,36,62,41,70,76,119,48,80,87,134,97,148,158,227,56,91,100,151,110,165,177,251,123,181,194,272,212,293,310,412,68,108,117,175,129,189,202,283,145,209,224,307,242,329,348,455,163,232,249,337,267,360,381,492,289,387,408,524,433,553,578,712,85,132,142,206,155,223,238,324,172,246,260,353,278,376,395,510,192,269,287,384,306,407,429,548,333,437,460,583,487,613,639,779,220,303,320,425,343,449,472,598,372,483,505,634,533,664,690,834,405,519,545,675,573,706,735,879,609,745,774,919,807,954,984,1133,25,46,51,84,58,94,102,154,66,106,114,171,126,185,198,277,78,121,131,191,144,208,222,305,161,230,245,332,263,356,375,486,92,141,152,219,166,237,252,342,182,259,274,371,294,394,414,532,205,286,302,404,323,428,448,572,352,459,482,608,509,638,663,806,112,168,179,254,195,273,290,388,215,297,314,417,336,441,464,588,241,328,346,453,370,481,502,631,399,514,537,668,565,699,725,870,270,365,385,497,409,525,550,681,438,560,585,720,614,752,781,927,475,602,625,766,655,799,824,975,694,838,865,1016,898,1049,1077,1227,140,204,218,301,236,322,341,447,258,351,369,480,393,508,530,661,285,383,403,518,427,547,571,705,458,581,607,743,637,777,805,952,317,422,442,566,469,595,619,759,501,630,654,798,686,830,856,1008,542,673,700,843,732,877,905,1055,771,916,946,1094,981,1130,1159,1303,364,474,496,624,523,653,679,822,559,693,718,863,751,897,925,1075,601,738,765,911,797,945,973,1123,837,987,1015,1165,1048,1199,1226,1369,648,791,817,967,849,1000,1028,1179,890,1041,1068,1219,1103,1251,1278,1419,939,1089,1117,1264,1151,1297,1324,1460,1193,1337,1363,1495,1395,1525,1548,1661,40,69,75,118,83,130,139,203,96,147,157,226,170,244,257,350,109,164,176,250,190,268,284,382,211,291,309,410,331,435,457,580,128,188,200,281,217,300,316,421,240,327,345,452,368,479,500,629,266,359,379,490,402,517,541,672,432,551,576,709,606,742,770,915,153,221,235,321,253,344,363,473,276,374,392,507,416,535,558,692,304,406,426,546,451,575,600,737,485,611,636,776,666,809,836,986,340,446,468,594,495,623,647,790,529,660,685,829,717,862,889,1040,570,704,731,876,764,910,938,1088,804,951,980,1129,1014,1164,1192,1336,187,265,280,378,299,401,420,540,325,430,450,574,477,604,627,768,358,465,489,615,516,644,671,813,549,680,707,850,740,884,913,1062,398,513,536,667,564,698,724,869,599,736,763,909,795,943,971,1121,642,782,810,957,842,991,1020,1169,880,1029,1058,1207,1092,1239,1267,1407,445,569,593,730,622,762,789,937,658,802,827,978,860,1012,1038,1190,703,845,875,1024,908,1057,1087,1235,949,1097,1127,1272,1162,1306,1334,1468,757,903,931,1081,965,1115,1143,1289,1005,1156,1183,1328,1216,1360,1385,1516,1054,1203,1231,1373,1263,1404,1429,1556,1301,1437,1463,1586,1493,1613,1634,1736,234,319,339,444,362,471,494,621,390,504,527,657,556,689,715,859,424,544,568,702, 597,734,761,907,633,773,801,948,833,983,1011,1161,467,592,617,756,646,788,815,964,683,826,853,1004,887,1037,1065,1215,729,874,902,1053,936,1086,1114,1262,977,1126,1155,1300,1189,1333,1359,1492,521,650,677,819,708,851,881,1030,748,893,922,1071,956,1105,1135,1280,793,941,969,1119,1002,1153,1181,1326,1044,1195,1222,1365,1253,1396,1421,1549,847,997,1026,1176,1059,1208,1236,1378,1100,1247,1275,1415,1308,1444,1470,1593,1148,1294,1321,1458,1352,1486,1509,1629,1391,1521,1544,1658,1571,1681,1700,1792,590,727,754,900,786,934,962,1112,823,974,1001,1152,1034,1186,1212,1356,872,1022,1051,1201,1084,1233,1260,1402,1124,1269,1298,1435,1331,1465,1490,1611,929,1079,1107,1255,1141,1287,1314,1451,1180,1325,1351,1485,1382,1513,1536,1651,1229,1371,1398,1527,1427,1554,1577,1686,1461,1584,1608,1712,1632,1734,1752,1833,995,1146,1174,1319,1206,1350,1377,1508,1244,1388,1412,1541,1442,1569,1591,1698,1292,1431,1456,1581,1484,1607,1628,1732,1519,1636,1656,1755,1680,1776,1791,1867,1346,1480,1504,1624,1532,1647,1667,1765,1563,1674,1693,1786,1717,1804,1819,1888,1604,1709,1729,1814,1749,1831,1846,1909,1772,1850,1864,1922,1878,1934,1943,1982,65,104,113,169,125,183,197,275,138,201,216,298,233,318,338,443,159,228,243,330,261,354,373,484,282,380,400,515,423,543,567,701,180,256,271,367,292,391,411,528,315,419,440,563,466,591,616,755,349,456,478,605,506,635,659,803,539,670,697,841,728,873,901,1052,214,295,313,415,335,439,463,586,361,470,493,620,520,649,676,818,396,511,534,665,562,696,722,867,596,733,760,906,792,940,968,1118,436,557,582,716,612,749,778,923,645,787,814,963,846,996,1025,1175,691,835,861,1013,895,1046,1073,1224,935,1085,1113,1261,1147,1293,1320,1457,255,347,366,476,389,503,526,656,418,538,561,695,589,726,753,899,454,577,603,739,632,772,800,947,669,811,839,988,871,1021,1050,1200,498,626,651,794,682,825,852,1003,721,866,894,1045,928,1078,1106,1254,767,912,942,1091,976,1125,1154,1299,1017,1166,1196,1339,1228,1370,1397,1526,555,688,714,858,747,892,921,1070,785,933,961,1111,994,1145,1173,1318,832,982,1010,1160,1043,1194,1221,1364,1083,1232,1259,1401,1291,1430,1455,1580,886,1036,1064,1214,1099,1246,1274,1414,1140,1286,1313,1450,1345,1479,1503,1623,1188,1332,1358,1491,1390,1520,1543,1657,1426,1553,1576,1685,1603,1708,1728,1813,311,413,434,554,461,584,610,746,491,618,643,784,674,816,844,993,531,662,687,831,719,864,891,1042,758,904,932,1082,966,1116,1144,1290,579,713,741,885,775,920,950,1098,812,960,990,1139,1023,1172,1202,1344,857,1009,1035,1187,1069,1220,1245,1389,1110,1258,1285,1425,1317,1454,1478,1602,640,780,808,955,840,989,1018,1167,878,1027,1056,1205,1090,1237,1265,1405,926,1076,1104,1252,1138,1284,1311,1448,1178,1323,1349,1483,1380,1511,1534,1649,985,1134,1163,1307,1197,1340,1367,1498,1234,1376,1403,1531,1432,1558,1582,1689,1279,1420,1443,1570,1473,1597,1617,1722,1507,1627,1646,1748,1669,1767,1782,1860,711,855,883,1033,918,1067,1096,1243,959,1109,1137,1283,1171,1316,1343,1477,1007,1158,1185,1330,1218,1362,1387,1518,1257,1400,1424,1552,1453,1579,1601,1707,1061,1211,1238,1381,1271,1411,1436,1562,1310,1447,1472,1596,1501,1621,1641,1743,1355,1489,1512,1631,1540,1655,1673,1771,1574,1684,170 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Zi(4096)：0,1,2,6,3,8,9,20,4,10,12,24,14,28,31,55,5,13,15,29,18,34,38,64,21,39,44，73,50,82,90,138,7,17,19,37,22,42,45,77,27,49,54,89,60,98,105,161,33,57,63,101,71,112,1 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Zi(8192)：0,1,2,6,3,8,9,20,4,10,12,24,14,28,31,55,5,13,15,29,18,34,38,64,21,39,44，73,50,82,90,138,7,17,19,37,22,42,45,77,27,49,54,89,60,98,105,161,33,57,63,101,71,112,121,181,81,125,137,201,150,219,235,329,11,23,26,47,30,53,59,95,35,61,67,107,74,117,128,190,43,72,79,123,88,136,147,216,99,151,163,237,177,256,274,381,52,86,93,144,103,157,169,246,116,175,188,271,204,291,311,428,134,198,212,301,231,325,345,470,254,354,377,508,404,542,572,746,16,32,36,62,41,70,76,120,48,80,87,135,97,149,159,233,56,91,100,152,111,167,180,259,124,185,199,284,218,308,327,447,68,109,118,178,130,194,207,296,146,215,230,324,249,349,370,501,165,239,257,358,278,385,409,547,304,418,443,590,474,628,661,851,85,133,143,211,156,229,245,343,174,253,269,375,290,402,426,569,197,281,300,413,323,442,468,621,353,479,506,667,540,708,743,949,226,319,339,463,364,493,522,687,398,535,564,737,599,778,817,1035,438,583,615,796,653,842,883,1111,702,900,942,1179,991,1236,1287,1572,25,46,51,84,58,94,102,155,66,106,115,173,127,189,203,289,78,122,132,196,145,214,228,322,162,236,252,352,272,379,401,539,92,142,153,225,168,244,260,363,186,268,286,397,309,425,449,598,210,299,318,437,342,467,492,652,374,505,534,701,568,742,777,990,113,170,183,263,200,285,305,419,221,312,332,453,357,484,513,675,248,348,368,499,396,533,561,734,430,575,604,785,642,830,869,1096,282,391,414,554,444,591,623,806,480,636,669,861,710,910,952,1191,526,693,725,930,768,980,1022,1273,822,1042,1086,1346,1142,1409,1463,1768,141,209,224,316,243,341,362,490,267,373,395,531,424,567,596,774,298,411,435,580,465,618,649,838,504,664,698,895,740,945,986,1230,336,459,485,643,519,683,716,918,560,732,767,978,813,1030,1074,1331,611,792,831,1051,878,1106,1152,1420,937,1173,1222,1498,1281,1565,1621,1943,390,525,553,723,589,766,804,1019,635,821,859,1083,908,1140,1188,1459,691,888,927,1163,977,1221,1269,1553,1040,1292,1342,1633,1406,1704,1763,2098,759,968,1010,1258,1064,1319,1372,1666,1131,1395,1449,1751,1515,1824,1888,2232,1210,1486,1540,1854,1608,1930,1993,2348,1692,2021,2085,2447,2164,2532,2604,2997,40,69,75,119,83,131,140,208,96,148,158,232,172,251,266,372,110,166,179,258,195,279,297,410,217,306,326,445,351,476,503,663,129,192,205,293,223,315,335,458,247,346,367,497,394,530,559,731,276,383,406,544,434,579,610,791,471,624,657,846,697,894,936,1172,154,227,242,340,261,365,388,523,288,400,423,566,451,601,633,819,320,439,464,617,495,655,689,886,537,704,739,944,781,994,1038,1290,361,489,518,682,551,721,757,966,595,773,812,1029,857,1081,1129,1393,648,837,877,11 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Zi(16384)：0,1,2,6,3,8,9,20,4,10,12,24,14,28,31,55,5,13,15,29,18,34,38,64,21,39,44，73,50,82,90,138,7,17,19,37,22,42,45,77,27,49,54,89,60,98,105,161,33,57,63,101,71,112,121,181,81,125,137,201,150,219,235,330,11,23,26,47,30,53,59,95,35,61,67,107,74,117,128,190,43,72,79,123,88,136,147,216,99,151,163,237,177,256,274,382,52,86,93,144,103,157,169,246,116,175,188,271,204,291,311,429,134,198,212,301,231,326,346,473,254,355,378,512,405,547,577,759,16,32,36,62,41,70,76,120,48,80,87,135,97,149,159,233,56,91,100,152,111,167,180,259,124,185,199,284,218,308,328,449,68,109,118,178,130,194,207,296,146,215,230,325,249,350,371,505,165,239,257,359,278,386,410,552,304,419,445,596,477,635,670,870,85,133,143,211,156,229,245,344,174,253,269,376,290,403,427,574,197,281,300,414,324,444,471,628,354,482,510,676,545,719,756,974,226,320,340,466,365,497,526,697,399,540,569,750,605,793,833,1067,440,589,622,812,662,861,904,1150,713,922,967,1224,1020,1287,1343,1663,25,46,51,84,58,94,102,155,66,106,115,173,127,189,203,289,78,122,132,196,145,214,228,323,162,236,252,353,272,380,402,544,92,142,153,225,168,244,260,364,186,268,286,398,309,426,451,604,210,299,319,439,343,470,496,661,375,509,539,712,573,755,792,1019,113,170,183,263,200,285,305,420,221,312,333,455,358,487,517,684,248,349,369,503,397,538,566,747,431,580,610,800,650,848,889,1134,282,392,415,559,446,597,630,822,483,643,678,880,721,932,977,1236,531,704,738,955,783,1009,1054,1329,839,1075,1123,1409,1184,1480,1540,1889,141,209,224,317,243,342,363,494,267,374,396,536,425,572,602,789,298,412,437,586,468,625,658,857,508,673,709,917,753,970,1015,1281,337,462,489,652,523,693,728,942,565,745,782,1007,829,1062,1110,1393,618,808,850,1086,899,1145,1196,1493,962,1218,1273,1581,1337,1656,1721,2095,391,529,558,735,595,780,820,1050,642,837,878,1119,930,1181,1233,1535,701,909,951,1207,1005,1271,1324,1643,1072,1348,1404,1733,1476,1815,1883,2278,773,995,1041,1312,1099,1379,1438,1771,1171,1463,1524,1869,1598,1954,2028,2439,1259,1567,1629,1991,1706,2080,2154,2583,1801,2186,2263,2704,2358,2810,2900,3406,40,69,75,119,83,131,140,208,96,148,158,232,172,251,266,373,110,166,179,258,195,279,297,411,217,306,327,447,352,479,507,672,129,192,205,293,223,316,336,461,247,347,368,501,395,535,564,744,276,384,407,549,436,585,617,807,474,631,666,865,708,916,961,1217,154,227,242,341,261,366,389,527,288,401,424,571,453,607,640,835,321,441,467,624,499,664,699,907,542,715,752,969,796,1023,1070,1346,362,493,522,692,556,733,771,993,601,788,828,1061,876,1117,1169,1461,657,856,898,1144,949,1205,1257,1565,1014,1280,1336,1655,1402,1731,1799,2184,191,275,292,406,315,435,459,615,345,472,498,663,533,706,740,957,383,518,548,722,584,766,805,1032,629,821,862,1100,914,1161,1213,1511,430,578,609,798,648,845,887,1131,698,905,948,1203,1001,1266,1320,1638,760,978,1025,1292,1081,1357,1417,1746,1151,1439,1500, 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Zi(32768)：0,1,2,6,3,8,9,20,4,10,12,24,14,28,31,55,5,13,15,29,18,34,38,64,21,39，44,73,50,82,90,138,7,17,19,37,22,42,45,77,27,49,54,89,60,98,105,161,33,57,63,101,71,112,121,181,81,125,137,201,150,219,235,330,11,23,26,47,30,53,59,95,35,61,67,107,74,117,128,190,43,72,79,123,88,136,147,216,99,151,163,237,177,256,274,382,52,86,93,144,103,157,169,246,116,175,188,271,204,291,311,429,134,198,212,301,231,326,346,473,254,355,378,512,405,547,578,761,16,32,36,62,41,70,76,120,48,80,87,135,97,149,159,233,56,91,100,152,111,167,180,259,124,185,199,284,218,308,328,449,68,109,118,178,130,194,207,296,146,215,230,325,249,350,371,505,165,239,257,359,278,386,410,553,304,419,445,597,477,636,671,875,85,133,143,211,156,229,245,344,174,253,269,376,290,403,427,575,197,281,300,414,324,444,471,629,354,482,510,677,545,720,758,980,226,320,340,466,365,497,526,698,399,540,570,752,606,796,837,1076,440,590,623,816,663,865,909,1162,714,927,973,1239,1027,1303,1362,1699,25,46,51,84,58,94,102,155,66,106,115,173,127,189,203,289,78,122,132,196,145,214,228,323,162,236,252,353,272,380,402,544,92,142,153,225,168, 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698,30211,30302,30741,29839,30338,30424,30850,30524,30937,31008,31354,30006,30483,30566,30973,30663,31056,31124,31453,30774,31150,31217,31530,31293,31594,31648,31893,27571,28299,28428,29077,28581,29215,29326,29883,28756,29373,29481,30020,29608,30133,30224,30675,28959,29554,29660,30177,29783,30287,30376,30806,29922,30410,30495,30911,30594,30997,31066,31403,29191,29763,29861,30358,29979,30462,30543,30955,30114,30579,30658,31053,30753,31135,31199,31517,30270,30712,30791,31165,30883,31244,31308,31607,30984,31332,31392,31679,31462,31738,31785,32007,29457,29999,30093,30561,30204,30657,30736,31120,30331,30769,30844,31213,30932,31289,31350,31645,30478,30896,30969,31319,31052,31391,31450,31728,31147,31472,31527,31793,31592,31847,31891,32095,30641,31039,31106,31439,31185,31505,31559,31821,31276,31582,31634,31883,31694,31933,31973,32163,31381,31669,31720,31953,31776,31999,32039,32215,31839,32052,32089,32256,32130,32289,32317,32438,28040,28727,28849,29456,28990,29583,29688,30203,29153,29729,29830,30330,29947,30433,30517,30931,29346,29899,29998,30477,30109,30574,30656,31051,30240,30687,30768,31146,30857,31223,31288,31591,29563,30092,30184,30640,30294,30735,30811,31184,30416,30843,30916,31275,31002,31349,31407,31693,30560,30968,31038,31380,31119,31449,31504,31775,31212,31526,31581,31838,31644,31890,31932,32129,29808,30310,30398,30826,30498,30914,30987,31335,30614,31015,31083,31418,31163,31486,31541,31805,30749,31131,31197,31515,31273,31579,31632,31881,31360,31653,31703,31940,31760,31988,32026,32206,30901,31261,31323,31621,31395,31682,31731,31963,31475,31750,31796,32017,31849,32061,32097,32263,31569,31829,31872,32081,31923,32122,32155,32311,31980,32168,32199,32346,32235,32374,32397,32503,30083,30552,30633,31032,30728,31113,31179,31500,30837,31207,31270,31577,31344,31640,31689,31929,30962,31314,31376,31666,31444,31724,31772,31997,31523,31790,31836,32050,31887,32092,32127,32287,31099,31433,31488,31762,31554,31817,31860,32071,31629,31879,31920,32120,31969,32160,32190,32339,31716,31950,31990,32176,32036,32213,32243,32380,32087,32254,32283,32411,32315,32436,32457,32547,31256,31564,31617,31868,31678,31919,31960,32152,31747,31977,32014,32196,32059,32233,32261,32395,31825,32042,32078,32248,32119,32282,32309,32433,32166,32319,32344,32460,32373,32484,32502,32583,31911,32112,32146,32303,32183,32333,32358,32472,32226,32367,32390,32497,32416,32517,32533,32607,32277,32407,32429,32527,32453,32545,32561,32628,32480,32565,32580,32641,32596,32653,32663,32702,25501,26387,26549,27359,26736,27532,27677,28393,26955,27736,27875,28577,28037,28724,28846,29453,27210,27968,28105,28785,28262,28926,29045,29632,28449,29094,29211,29779,29343,29896,29995,30474,27506,28238,28369,29023,28523,29160,29274,29835,28700,29322,29430,29975,29560,30089,30181,30637,28906,29507,29613,30136,29735,30245,30334,30770,29879,30372,30458,30879,30557,30965,31035,31377,27844,28547,28674,29296,28818,29427,29536,30066,28988,29581,29686,30201,29806,30308,30396,30824,29182,29754,29855,30352,29971,30454,30537,30949,30107,30572, 30654,31049,30747,31129,31195,31513,29408,29953,30049,30521,30161,30618,30698,31086,30292,30733,30809,31182,30899,31259,31321,31619,30438,30862,30934,31290,31019,31364,31421,31705,31117,31447,31502,31773,31567,31827,31870,32079,28228,28897,29015,29606,29152,29728,29829,30329,29312,29870,29967,30451,30081,30550,30631,31030,29499,30034,30131,30592,30239,30686,30767,31145,30365,30797,30874,31237,30960,31312,31374,31664,29708,30222,30311,30751,30415,30842,30915,31274,30533,30946,31016,31362,31097,31431,31487,31761,30673,31064,31133,31460,31211,31525,31580,31837,31301,31601,31655,31899,31714,31948,31989,32175,29946,30432,30516,30930,30613,31014,31082,31417,30727,31112,31178,31499,31255,31563,31616,31867,30856,31222,31287,31590,31359,31652,31702,31939,31443,31723,31771,31996,31824,32041,32077,32247,31001,31348,31406,31692,31474,31749,31795,32016,31553,31816,31859,32070,31910,32111,32145,32302,31643,31889,31931,32128,31979,32167,32198,32345,32035,32212,32242,32379,32276,32406,32428,32526,28662,29286,29399,29945,29526,30058,30154,30612,29676,30192,30285,30726,30388,30817,30894,31254,29846,30343,30431,30855,30529,30941,31013,31358,30646,31042,31111,31442,31188,31507,31562,31823,30042,30515,30597,31000,30692,31081,31148,31473,30804,31177,31242,31552,31317,31615,31667,31909,30929,31286,31347,31642,31416,31701,31748,31978,31498,31770,31815,32034,31866,32076,32110,32275,30263,30707,30787,31162,30877,31240,31304,31604,30979,31328,31390,31677,31458,31735,31783,32005,31094,31428,31485,31759,31550,31813,31857,32068,31625,31875,31918,32118,31966,32157,32188,32337,31228,31540,31593,31848,31657,31901,31942,32137,31727,31959,31998,32182,32044,32219,32249,32384,31804,32025,32060,32234,32103,32269,32295,32422,32151,32308,32332,32452,32361,32475,32492,32576,30508,30923,30996,31343,31076,31412,31471,31746,31171,31494,31548,31812,31611,31864,31907,32109,31281,31585,31639,31886,31697,31935,31976,32165,31767,31993,32032,32210,32074,32245,32274,32405,31402,31688,31737,31968,31792,32013,32051,32225,31855,32067,32102,32268,32142,32300,32326,32446,31928,32126,32159,32314,32195,32343,32366,32479,32240,32378,32401,32506,32426,32525,32540,32613,31535,31800,31846,32058,31897,32100,32135,32293,31956,32149,32181,32331,32217,32360,32383,32491,32021,32202,32232,32372,32266,32399,32420,32520,32306,32431,32451,32544,32474,32562,32575,32638,32094,32260,32288,32415,32320,32440,32461,32550,32356,32470,32488,32572,32509,32588,32601,32657,32394,32501,32516,32595,32535,32609,32619,32671,32559,32627,32636,32684,32648,32692,32698,32727,29138,29715,29817,30318,29934,30421,30506,30921,30069,30540,30621,31022,30716,31103,31169,31492,30228,30677,30757,31137,30847,31214,31279,31583,30952,31305,31367,31658,31436,31717,31765,31991,30406,30833,30907,31266,30994,31341,31400,31686,31089,31424,31480,31755,31546,31810,31853,32065,31203,31519,31573,31832,31637,31884,31926,32124,31708,31943,31984,32171,32030,32208,32238,32376,30604,31006,31075,31411,31154,31478,31534,31799,31247,31557,31610,31863,31671,31913,31955,321 48,31352,31646,31696,31934,31753,31982,32020,32201,31819,32037,32073,32244,32114,32278,32305,32430,31470,31745,31791,32012,31845,32057,32093,32259,31906,32108,32141,32299,32180,32330,32355,32469,31975,32164,32194,32342,32231,32371,32393,32500,32273,32404,32425,32524,32450,32543,32558,32626,30827,31198,31262,31570,31336,31633,31683,31924,31419,31704,31751,31981,31806,32027,32062,32236,31516,31784,31830,32045,31882,32088,32123,32284,31941,32136,32169,32321,32207,32352,32375,32485,31622,31873,31915,32116,31964,32156,32186,32336,32018,32200,32229,32370,32264,32398,32418,32519,32082,32250,32280,32409,32312,32434,32455,32546,32347,32462,32482,32567,32504,32584,32597,32654,31741,31972,32010,32193,32054,32228,32258,32392,32105,32271,32297,32424,32328,32448,32468,32557,32162,32316,32341,32458,32369,32481,32499,32581,32403,32507,32523,32599,32542,32614,32625,32675,32223,32365,32387,32495,32414,32515,32531,32605,32444,32539,32553,32622,32571,32635,32645,32689,32478,32564,32578,32640,32594,32652,32661,32701,32612,32665,32673,32709,32683,32716,32721,32742,31068,31405,31464,31740,31528,31794,31840,32053,31605,31858,31902,32104,31951,32144,32177,32327,31691,31930,31971,32161,32015,32197,32227,32368,32069,32241,32270,32402,32301,32427,32447,32541,31787,32009,32047,32222,32090,32257,32285,32413,32138,32296,32323,32443,32353,32467,32486,32570,32192,32340,32364,32477,32391,32498,32514,32593,32423,32522,32538,32611,32556,32624,32634,32682,31892,32096,32131,32290,32170,32322,32348,32463,32214,32357,32381,32489,32408,32510,32528,32602,32262,32396,32417,32518,32442,32537,32552,32621,32471,32560,32573,32637,32589,32649,32658,32699,32318,32439,32459,32549,32483,32568,32582,32643,32508,32587,32600,32656,32615,32667,32676,32711,32534,32608,32618,32670,32632,32680,32687,32719,32647,32691,32697,32726,32705,32731,32735,32751,32006,32189,32220,32362,32255,32389,32412,32513,32294,32421,32441,32536,32466,32555,32569,32633,32338,32456,32476,32563,32496,32579,32592,32651,32521,32598,32610,32664,32623,32674,32681,32715,32385,32493,32511,32590,32530,32604,32616,32668,32551,32620,32631,32679,32644,32688,32695,32724,32577,32639,32650,32693,32660,32700,32706,32732,32672,32708,32714,32737,32720,32741,32744,32756,32437,32532,32548,32617,32566,32630,32642,32686,32586,32646,32655,32696,32666,32704,32710,32734,32606,32662,32669,32707,32678,32713,32718,32740,32690,32722,32725,32745,32730,32748,32750,32760,32629,32677,32685,32717,32694,32723,32728,32746,32703,32729,32733,32749,32738,32752,32754,32762,32712,32736,32739,32753,32743,32755,32757,32763,32747,32758,32759,32764,32761,32765,32766,32767；
(二)、子信道行重序列RW：
RW_8：1,2,2,4,2,4,4,8；
RW_16：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16；
RW_32：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32；
RW_64：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4,4，8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64；
RW_128：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4,4,8，4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128；
RW_256：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4,4,8，4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256；
RW_512：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4,4,8，4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512；
RW_1024：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4,4，8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32, 16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024；
RW_2048：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4,4，8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,2,4,4,8,4,8,8, 16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64, 64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048；
RW_4096：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4，4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,2,4,4,8,4,8, 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256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096；
RW_8192：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4,4，8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32, 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128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192；
RW_16384：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4，4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,2,4,4,8,4,8,8,16,4,8,8, 16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,1 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256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,1024,2048,2048,4096,2048,4096,4096,8192,2048,4096,4096,8192,4096,8192,8192,16384；
RW_32768：1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,2,4，4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128, 8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,256,64,128,128,256,128,256,256,512,8,16,16,32,16,32,32,64,16,32,32,64,32,64,64,128,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,16,32,32,64,32,64,64,128,32,64,64,128,64,128,128,256,32,64,64,128,64,128,128,25 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024,1024,2048,1024,2048,2048,4096,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,1024,2048,2048,4096,2048,4096,4096,8192,2048,4096,4096,8192,4096,8192,8192,16384,64,128,128,256,128,256,256,512,128,256,256,512,256,512,512,1024,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,1024,2048,2048,4096,2048,4096,4096,8192,2048,4096,4096,8192,4096,8192,8192,16384,128,256,256,512,256,512,512,1024,256,512,512,1024,512,1024,1024,2048,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,1024,2048,2048,4096,2048,4096,4096,8192,2048,4096,4096,8192,4096,8192,8192,16384,256,512,512,1024,512,1024,1024,2048,512,1024,1024,2048,1024,2048,2048,4096,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,512,1024,1024,2048,1024,2048,2048,4096,1024,2048,2048,4096,2048,4096,4096,8192,1024,2048,2048,4096,2048,4096,4096,8192,2048,4096,4096,8192,4096,8192,8192,16384,512,1024,1024,2048,1024,2048,2048,4096, 1024,2048,2048,4096,2048,4096,4096,8192,1024,2048,2048,4096,2048,4096,4096,8192,2048,4096,4096,8192,4096,8192,8192,16384,1024,2048,2048,4096,2048,4096,4096,8192,2048,4096,4096,8192,4096,8192,8192,16384,2048,4096,4096,8192,4096,8192,8192,16384,4096,8192,8192,16384,8192,16384,16384,32768；
(三)、汉明重量序列HW：
HW_8：0,1,1,2,1,2,2,3；
HW_16：0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4；
HW_32：0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5；
HW_64：0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,3,2,3,3,4，2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6；
HW_128：0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,3,2,3,3，4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7；
HW_256：0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,3,2,3,3，4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8；
HW_512：0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,3,2,3,3，4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9；
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HW_2048：0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,3,2,3,3，4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4, 5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11；
HW_4096：0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,2,3,2,3,3，4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6, 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7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10, 10,11,10,11,11,12,10,11,11,12,11,12,12,13,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,10,11,11,12,11,12,12,13,11,12,12,13,12,13,13,14,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,10,11,11,12,11,12,12,13,11,12,12,13,12,13,13,14,5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9, 10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,10,11,11,12,11,12,12,13,11,12,12,13,12,13,13,14,6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,10,11,11,12,11,12,12,13,11,12,12,13,12,13,13,14,7,8,8,9,8,9,9,10,8,9,9,10,9,10,10,11,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,10,11,11,12,11,12,12,13,11,12,12,13,12,13,13,14,8,9,9,10,9,10,10,11,9,10,10,11,10,11,11,12,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,10,11,11,12,11,12,12,13,11,12,12,13,12,13,13,14,9,10,10,11,10,11,11,12,10,11,11,12,11,12,12,13,10,11,11,12,11,12,12,13,11,12,12,13,12,13,13,14,10,11,11,12,11,12,12,13,11,12,12,13,12,13,13,14,11,12,12,13,12,13,13,14,12,13,13,14,13,14,14,15。
三、校验方程
无论是采用哪种编码方式，例如，CA-Polar、PC-Polar或者CA+PC-Polar编码方法，其校验方程都可以由生成多项式唯一确定，而生成多项式又有多种等价的表示方式，例如可以采用二进制，八进制，十进制，十六进制，位置向量中的任一种进行表示。例如：若CRC多项式为[1,1,0,0,1,1,1],其也可以等价表示为其1的位置向量(最低位从0开始)[6,5,2,1,0]；也可以由其2进制向量对应的十进制数表示，即，下文给出的十进制表示为将二进制向量高低位翻转后，转换成的十进制值，如对于多项式为[1,1,0,0,1,1,1],其十进制表示为115。
(1)对于在编码过程中采用CRC比特进行校验的编码方式，校验方程可以生成多项式(polynomial产生)，例如，假设生成多项式采用十进制表示，则这些生成多项式可以是：(24):29254851；(24):25952259；(19):1005307；(16):67601；(15):57977；(14):31993；(13):16299；(12):7705；(11):3805；(10):1939；(9):817；(8):435；(7):211；(6):115；(5):41；(4):25；(3):9；
上述括号中的数值表示CRC的长度(即CRC比特数)，括号后的序列表示多项式。针对每种CRC长度，多项式也可以是其它的序列，上述例子只是用于帮助理解。在上述例子中，多项式是采用十进制表示的，在实际应用中还可以采用其它进制表示，对此本申请不做限制。
(2)对于在编码过程中采用PC比特进行校验的编码方式，校验方程还可以采用表格表示法：图6为校验方程的表格表示法的示意图，如图6所示冻结/信息/校验/缩短位置分别由不同填充表示，表格中的F表示固定比特，I表示信息比特，S表示缩短比特，其他的表格中的数字代表其参与的第几个校验方程，相同数字对应位置的比特参与同一个校验方程，其中子信道序号最大(最靠后)的比特为PC比特。
下面列举几种表格表示的校验方式的实例：
Figure BDA0001220174590002451
除了上述的使用多种进制的数字表示多项式，来确定校验方程，或者使用表格表示校验方程之外，还可以通过另一种多项式表示校验方程，即可以使用移位寄存器进行表示。
PC-Polar校验方程的多项式表示法：作为更为简便的校验方程表示方式，每个校验方程只需要1个向量，或一个十进制数表示即可。下面举几个移位寄存器表示多项式的实例。
图7为多项式[1,0,0,0,0]采用移位寄存器的示意图，如图7所示，对于长度为5的移位寄存器，其多项式表示可以为[1,0,0,0,0]或者[4]或者十进制数16。
图8为多项式[1,0,1,0,0]采用移位寄存器的示意图，如图8所示，对于长度为5，但在第3个位置具有反馈抽头的的移位寄存器，其多项式表示可以为[1,0,1,0,0]或者[4,2]或者十进制数20。
此外，对于PC校验方程，可用(但不仅限于)的多项式的十进制表示为：4；16；64；256；272；68；273；65552；65808；65553；70661；1028；4100；65540；1029；65604；21；69633；69636；69648；69636。作为一种可选的方式，PC-Polar也可以使用CRC多项式直接作为其校验方程的多项式。
需要说明的是，对于CA+PC-Polar编码方式：(1)PC比特和CRC比特可以使用不同的多项式来定义校验方程，且相互独立进行预编码(例如，采用两组不同的移位寄存器)；(2)PC比特和CRC比特使用相同的多项式来定义校验方程，且同时联合进行预编码(例如，采用一组相同的移位寄存器)；(3)PC比特和CRC比特采用不同的校验方程，例如，PC比特使用表格的方式来定义非多项式的校验方程，CRC比特使用移位寄存器来定义校验方程。
如果发送端采用CA-Polar或者CA+PC-Polar的编码方式，则接收端对应的可以采用CA+PC-SCL的译码方式。由于CA+PC-Polar的编码方式中采用了校验比特(例如，CRC比特)进行校验，正常情况下，接收端在译码过程中需要对list(列表)中所有的译码路径(例如L条译码路径)都使用CRC进行检验，以确定被检查的路径是否正确。但是本申请人的发明人发现，每次CRC检验都会增加一定的虚警概率。“虚警”是指：译码错误但是使用CRC比特检验通过。这是因为可能存在多个译码结果都具有相同的CRC校验值，但其中只有唯一一个是正确的译码结果。
由此可见，虚警概率和使用CRC比特进行校验的次数密切相关。基于此，本申请的实施例还提供了一种虚警概率可控的译码方法，图9为本申请的译码方法的译码过程示意图，如图9所示，通过此方法，可以不用对每条译码路径都进行CRC检验，由此提高了译码的效率。采用采用CA+PC-Polar编码方式编码后的序列中包括四种比特，即：信息比特、静态冻结比特(或称为冻结比特，此比特为可选，也就是说，在编码后的序列中也有可能不包括静态冻结比特)、PC比特和CRC比特。静态冻结比特(或称为冻结比特，此比特为可选，也就是说，在编码后的序列中也有可能不包括静态冻结比特)
接收端因为知道每种比特的类型、数量以及对应的位置，因此在译码时也会区分对待。假设，发送端经过编码后输出的编码块中包含N个比特，接收端采用的译码列表的宽度为L，接收端收到每个比特后都会根据每个比特的类型进行不同的处理：
(1)对于信息比特，则将该信息比特对应的路径扩展为0和1的两个路径，并 分别计算两条路径的度量值,即PM值；
如果扩展后的路径数l≤L，则保留所有路径；
如扩展后路径数l>L，则保留PM值最优(最小)的L条路径；
(2)对于静态冻结比特，则不扩展路径，直接将该比特译码结果置为0，并计算此静态冻结比特所在路径的PM值；
(3)对于PC比特或者CRC比特，在处理时可以采用如下两种方式中的任意一种：
a.方法一：不扩展路径，直接将该比特的值置为满足校验方程的值，并计算相应的PM值；
如果是CRC比特，则在译到最后一个CRC比特之前，可以按照上述方法一进行处理，如果译到最后一个CRC比特时，可以按照下面的方法二的第ii中方式进行处理。
b.方法二：
i.在译码过程中，例如，译到最后一个CRC比特之前，按照与信息比特完全相同的方法扩展路径，并计算每条路径的PM值；
ii.在译码结束时，例如，译到最后一个CRC比特时，按PM值从好到差(从小到大)依次做最多T次CRC检验，其中T满足1≤T≤L，例如，参见图9所示，列表的宽度L＝16，其中只进行了T＝4次CRC检验。
ii.1若检验次数≤T时，已经找到一条校验通过的路径，则译码成功，并返回CRC通过的那条路径作为译码结果；
ii.2.若检验次数>T时，仍未找到一条校验通过的路径，则译码失败。
在前面的实施例中已经介绍过编码过程中需要使用到的参数，即：校验比特的数量、校验比特的位置，校验方程，接收端在译码的过程中同样需要使用到这些参数。如果是采用上述虚警概率可控的CA+PC-SCL译码方法的话，则还需要进一步知道允许CRC检验的最大次数T的值。
在一个可选的实施例中，为了满足虚警概率可控，且虚警概率与译码器列表宽度(L的大小)无关，在基于前面描述的确定CRC比特数量的方法的基础上，还可以使得CRC比特数量J、T和最大虚警概率FAR(false alarm rate)满足如下关系：
FAR≈2-J×T
或者所述CRC比特长度至满足：
J≥log2(T/FAR)
下面通过两个具体的例子来进行说明：
例1，给定单次译码的虚警概率上限为FAR＝10-5，而当前译码器参数T＝4，则所需的CRC长度至少为J＝ceil(log2(4/10-5))＝19比特。
例2，如在T＝1时，J＝16比特已能满足虚警概率不超过设定值FAR，则T＝8时，若仍要满足虚警概率不超过原设定值FAR，则所需的额外CRC比特数应满足ΔJ＝log2(T)，即
J(T)＝J(1)+ΔJ＝J(1)+log2T
上式中J(1)为单次CRC检测时满足要求的CRC比特个数，J(T)为T次CRC检测是满足要求的CRC比特个数。
在一个可选的实施例中，上述J和T的值有可能是通过协议预先规定的。
(1)如果单次校验最少需要8个CRC比特，则，J和T的值可以是如下表中的任意一种组合：
组合1 组合2 组合3 组合4 组合5 组合6 组合7 组合8 组合9
J的值 8 9 10 11 12 13 14 15 16
T的值 1 2 4 8 16 32 64 128 256
(2)如果单次校验最少需要16个CRC比特，则J和T的值可以是如下表中的任意一种组合：
组合1 组合2 组合3 组合4 组合5 组合6 组合7 组合8 组合9
J的值 16 17 18 19 20 21 22 23 24
T的值 1 2 4 8 16 32 64 128 256
(3)如果单次校验最少需要24个CRC比特，则J和T的值可以是如下表中的任意一种组合：
组合1 组合2 组合3 组合4 组合5 组合6 组合7 组合8 组合9
J的值 24 25 26 27 28 29 30 31 32
T的值 1 2 4 8 16 32 64 128 256
(4)如果单次校验最少需要32个CRC比特，则，J和T的值可以是如下表中的任意一种组合：
组合1 组合2 组合3 组合4 组合5 组合6 组合7 组合8 组合9
J的值 32 33 34 35 36 37 38 39 40
T的值 1 2 4 8 16 32 64 128 256
在一个可选的实施例中，为了使在各种列表宽度L下，都具有较好的误块率(BLER)性能，需要设定一定数量的A类PC比特，例如，PC比特的数量P的值有可能是如下等号后面所列的多个值中的任意一个：
P＝1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16。
T和P的取值与编码端采用的编码方式息息相关，例如：
情况(1)：如果T＝L,P＝0,则编码端采用的是CA-Polar编码方式，其虚警概率不可控；
情况(2):如果T＝1,P>0,则编码端采用的是PC-Polar编码方式，其虚警概率可控；
情况(3):如果1<T<L且T为不随L变化的常数，且P>0,则编码端采用的是PC+CA-Polar编码方式，其虚警概率可控；
情况(4):如果1<T<L且T为不随L变化的常数，且P＝0,所构造码即为虚警概率可控的CA-Polar码。
在上述方案中，应理解，T＝1时，T次校验指的是该路径对应的译码结果的一次检错。
由于每次检错(CRC检查)都会增加一定的虚警概率，“虚警”的定义为”译码错误但未检测出译码错误(如CRC通过)”，这是因为存在多个译码结果具有相同的检错校验值(如CRC校验值)，但其中只有唯一一个正确译码结果。因此，虚警概率随检错次数(如CRC使用次数)成比例上升。对此，本申请提出的方案限制了检错次数(如CRC使用次数)，从根本上解决了虚警概率不可控的问题。在传统方法中，列表译码的List越大，则虚警概率越高；而在本申请提出的方案中，虚警概率不随list增大而增大。
图10为本申请提供的信息的传输装置实施例的结构示意图；如图10所示，该信息的传输装置10包括：
处理模块11，用于根据预设的参数对的待编码的序列进行编码，得到编码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程；所述待编码的序列长度为K，所述K为大于0的整数；
发送模块12，用于将所述编码后的序列发送至接收设备。
关于校验比特的数量、校验比特的位置以及校验方程的配置或者确定可以参见前面方法实施例中的描述，此处不再赘述。
在实现时上述处理模块11可以是一个或多个处理器，当处理器中存储有程序代码时，该处理器运行该程序代码就可以实现上述处理模块11的功能，当处理器中没有存储有程序代码时，可以通过通过读取外部存储器中存储的程序代码来实现上述处理模块11的功能。上述处理模块11还可以处理电路。上述发送模块12可以是一个发送器或者发送电路。
本实施例中提供的信息的传输装置，可以用于执行前述任一方法实施例中发送设备侧的技术方案，其实现原理和技术效果与方法实施例类似，在此不再赘述。
本实施例还提供了一种编码装置，该装置包括上述处理模块11、输入接口和输出接口；输入接口用于将待编码的序列输入给处理模块11，输出接口用于输出经过处理模块11编码后的序列。
本实施例中提供的编码装置，可以是通过芯片或者电路实现，其用于执行前述任一方法实施例中发送设备侧的编码技术方案，因实现原理和技术效果与方法实施例类似，在此不再赘述。
图11为本申请提供的信息的传输装置实施例二的结构示意图；如图11所示，该信息的传输装置20包括：
接收模块21，用于接收发送设备发送的编码后的序列；所述编码后的序列是根据预设的参数对待编码的序列进行编码后得到的，所述待编码的序列长度为K，所述K为大于0的整数；
处理模块22，用于根据所述预设的参数对所述编码后的序列进行译码，得到译码后 的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程。
关于校验比特的数量、校验比特的位置以及校验方程的配置或者确定可以参见前面方法实施例中的描述，此处不再赘述。
本实施例提供的信息的传输装置，用于执行前述任一方法实施例中接收设备侧的技术方案，其实现原理和技术效果类似，在此不再赘述。
本实施例提供的信息的传输装置20，还可包括输入接口和输出接口；输入接口用于将编码后的序列输入给处理模块22，输出接口用于输出经过处理模块22译码后的序列。
本实施例中提供的信息的传输装置，可以是通过芯片或者电路实现，其用于执行前述任一方法实施例中接收设备侧的译码技术方案，因实现原理和技术效果与方法实施例类似，在此不再赘述。
图12为本申请提供的译码装置实施例的结构示意图；如图12所示，该译码装置30包括：
接收模块31，用于从发送设备接收待译码的比特序列；所述待译码的比特序列中至少包括信息比特和至少一个校验比特；
处理模块32，用于采用串行抵消列表SCL译码算法按顺序对所述待译码的比特进行译码，输出度量值最优的l条候选路径；所述1≤l≤L，所述L为SCL的宽度，所述L为大于或等于1的整数；
所述处理模块32还用于从度量值最优的候选路径开始，用所述至少一个校验比特对其中的T条候选路径进行校验，将校验通过的第一个候选路径中的比特序列作为译码输出；所述T为整数，所述1≤T≤l。
本实施例提供的译码装置，还可包括输入接口和输出接口；输入接口用于将编码后的序列输入给处理模块32，输出接口用于输出经过处理模块32译码后的序列。
本实施例中提供的译码装置，可以是通过芯片或者电路实现，其用于执行前述任一方法实施例中接收设备侧的译码技术方案，因实现原理和技术效果与方法实施例类似，在此不再赘述。
本申请还提供一种可读存储介质，可读存储介质中存储有指令，当发送设备的至少一个处理器执行该指令时，发送设备执行上述任一方法实施例中提供的信息的传输方法以及编码方法。
本申请还提供一种可读存储介质，可读存储介质中存储有指令，当接收设备的至少一个处理器执行该指令时，接收设备执行上述任一方法实施例中提供的信息的传输方法以及译码方法。
本申请还提供一种程序产品，该程序产品包括指令，该指令存储在可读存储介质中。发送设备的至少一个处理器可以从可读存储介质读取该指令，并执行该指令使得发送设备实施任一方法实施例中提供的信息的传输方法以及编码方法。
本申请还提供一种程序产品，该程序产品包括指令，该指令存储在可读存储介质中。接收设备的至少一个处理器可以从可读存储介质读取该指令，并执行该指令使得接收设备实施上述任一方法实施例中提供的信息的传输方法以及译码方法。
在发送设备或者接收设备的具体实现中，应理解，处理器可以是中央处理单元(英文：Central Processing Unit，简称：CPU)，还可以是其他通用处理器、数字信号处理器(英文：Digital Signal Processor，简称：DSP)、专用集成电路(英文：Application SpecificIntegrated Circuit，简称：ASIC)等。通用处理器可以是微处理器或者该处理器也可以是任何常规的处理器等。结合本申请所公开的方法的步骤可以直接体现为硬件处理器执行完成，或者用处理器中的硬件及软件模块组合执行完成。
实现上述各方法实施例的全部或部分步骤可以通过程序指令相关的硬件来完成。前述的程序可以存储于一可读取存储器中。该程序在执行时，执行包括上述各方法实施例的步骤；而前述的存储器(存储介质)包括：只读存储器(英文：read-only memory，缩写：ROM)、RAM、快闪存储器、硬盘、固态硬盘、磁带(英文：magnetic tape)、软盘(英文：floppydisk)、光盘(英文：optical disc)及其任意组合。
最后应说明的是：尽管参照前述各实施例对本方案进行了详细的说明，本领域的普通技术人员应当理解：其依然可以对前述各实施例所记载的技术方案进行修改，或者对其中部分或者全部技术特征进行等同替换；而这些修改或者替换，并不能使相应技术方案的本质脱离本申请各实施例技术方案的范围。

Claims (26)

1.一种信息的传输方法，其特征在于，包括：
发送设备根据预设的参数对待编码的序列进行编码，得到编码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程；所述待编码的序列长度为K，所述K为大于0的整数；
所述发送设备将所述编码后的序列发送至接收设备；
所述校验比特的位置是根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
2.根据权利要求1所述的方法，其特征在于：
所述校验比特的数量是预先配置的；
或者，
所述校验比特的数量是根据译码器的列表长度确定的；
或者，
所述校验比特的数量是根据码长和码率确定的；
或者，
所述校验比特的数量是根据预先配置的比特数、译码器的列表长度、码长和码率确定的。
3.根据权利要求1或2所述的方法，其特征在于，所述校验方程是根据多项式确定的；其中，所述多项式采用以下任一种形式表示：二进制，八进制，十进制，十六进制，位置向量。
4.根据权利要求1或2所述的方法，其特征在于，所述校验方程是根据校验方程表格确定的；其中，所述校验方程表格至少包括校验比特、信息比特以及参与校验方程的信息比特。
5.一种信息的传输方法，其特征在于，包括：
接收设备接收发送设备发送的编码后的序列；所述编码后的序列是根据预设的参数对待编码的序列进行编码后得到的，所述待编码的序列长度为K，所述K为大于0的整数；
所述接收设备根据所述预设的参数对所述编码后的序列进行译码，得到译码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程；
所述校验比特的位置是根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
6.根据权利要求5所述的方法，其特征在于：
所述校验比特的数量是预先配置的；
或者，
所述校验比特的数量是根据译码器的列表长度确定的；
或者，
所述校验比特的数量是根据码长和码率确定的；
或者，
所述校验比特的数量是根据预先配置的比特数、译码器的列表长度、码长和码率确定的。
7.根据权利要求5或6所述的方法，其特征在于，所述校验方程是根据多项式确定的；其中，所述多项式采用以下任一种形式表示：二进制，八进制，十进制，十六进制，位置向量。
8.根据权利要求5或6所述的方法，其特征在于，所述校验方程是根据校验方程表格确定的；其中，所述校验方程表格至少包括校验比特、信息比特以及参与校验方程的信息比特。
9.一种信息的传输装置，其特征在于，包括：
处理模块，用于根据预设的参数对待编码的序列进行编码，得到编码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程；所述待编码的序列长度为K，所述K为大于0的整数；
发送模块，用于将所述编码后的序列发送至接收设备；
所述校验比特的位置是根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
10.根据权利要求9所述的装置，其特征在于：
所述校验比特的数量是预先配置的；
或者，
所述校验比特的数量是根据译码器的列表长度确定的；
或者，
所述校验比特的数量是根据码长和码率确定的；
或者，
所述校验比特的数量是根据预先配置的比特数、译码器的列表长度、码长和码率确定的。
11.根据权利要求9或10所述的装置，其特征在于，所述校验方程是根据多项式确定的；其中，所述多项式采用以下任一种形式表示：二进制，八进制，十进制，十六进制，位置向量。
12.根据权利要求9或10所述的装置，其特征在于，所述校验方程是根据校验方程表格确定的；其中，所述校验方程表格至少包括校验比特、信息比特以及参与校验方程的信息比特。
13.一种信息的传输装置，其特征在于，包括：
接收模块，用于接收发送设备发送的编码后的序列；所述编码后的序列是根据预设的参数对待编码的序列进行编码后得到的，所述待编码的序列长度为K，所述K为大于0的整数；
处理模块，用于根据所述预设的参数对所述编码后的序列进行译码，得到译码后的序列；所述参数包括校验比特的数量、所述校验比特的位置以及校验方程；
所述校验比特的位置是根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
14.根据权利要求13所述的装置，其特征在于：
所述校验比特的数量是预先配置的；
或者，
所述校验比特的数量是根据译码器的列表长度确定的；
或者，
所述校验比特的数量是根据码长和码率确定的；
或者，
所述校验比特的数量是根据预先配置的比特数、译码器的列表长度、码长和码率确定的。
15.根据权利要求13或14所述的装置，其特征在于，所述校验方程是根据多项式确定的；其中，所述多项式采用以下任一种形式表示：二进制，八进制，十进制，十六进制，位置向量。
16.根据权利要求13或14所述的装置，其特征在于，所述校验方程是根据校验方程表格确定的；其中，所述校验方程表格至少包括校验比特、信息比特以及参与校验方程的信息比特。
17.一种译码方法，其特征在于，所述方法包括：
接收设备接收待译码的比特序列；所述待译码的比特序列中至少包括信息比特和至少一个校验比特；所述接收设备采用串行抵消列表SCL译码算法按顺序对所述待译码的比特进行译码，输出度量值最优的l条候选路径；所述1≤l≤L，所述L为SCL的宽度，所述L为大于或等于1的整数；
从度量值最优的候选路径开始，用所述至少一个校验比特对其中的T条候选路径进行校验，将校验通过的第一个候选路径中的比特序列作为译码输出；所述T为整数，所述1≤T≤l；
所述校验比特的位置是根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
18.根据权利要求17所述的方法，其特征在于，所述接收设备采用串行抵消列表SCL译码算法按顺序对所述待译码的比特进行译码包括：
对于所述信息比特，则将该信息比特对应的路径扩展为0和1的两个路径，并分别计算两条路径的度量值；
如果扩展后的路径数小于或等于L，则保留所有路径；
如扩展后路径数大于L，则保留度量值最优的L条路径。
19.根据权利要求17或18所述的方法，其特征在于，所述接收设备采用串行抵消列表SCL译码算法按顺序对所述待译码的比特进行译码包括：
所述校验比特有多个，当所述接收设备译到最后一个校验比特之前，则将该最后一个校验比特之前的校验比特对应的路径扩展为0和1的两个路径，并分别计算两条路径的度量值；
如果扩展后的路径数小于或等于L，则保留所有路径；
如扩展后路径数大于L，则保留度量值最小的L条路径；
所述从度量值最优的候选路径开始，对其中的T条候选路径进行校验包括：
当所述接收设备译到最后一个校验比特时，所述接收设备从保留的l条路径中度量值最优的候选路径开始，使用所述最后一个校验比特对其中的T条候选路径进行校验。
20.根据权利要求17或18任一项所述的方法，其特征在于，所述待译码的比特序列中还包括校验冻结比特，所述校验冻结比特的数量P为如下多个值中的一个：
P＝1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16。
21.一种译码装置，其特征在于，包括：
接收模块，用于接收待译码的比特序列；所述待译码的比特序列中至少包括信息比特和至少一个校验比特；
处理模块，用于采用串行抵消列表SCL译码算法按顺序对所述待译码的比特进行译码，输出度量值最优的l条候选路径；所述1≤l≤L，所述L为SCL的宽度，所述L为大于或等于1的整数；
所述处理模块还用于从度量值最优的候选路径开始，用所述至少一个校验比特对其中的T条候选路径进行校验，将校验通过的第一个候选路径中的比特序列作为译码输出；所述T为整数，所述1≤T≤l；
所述校验比特的位置是根据位置参数确定的，所述位置参数至少包括信息阈值ZI和校验阈值ZP；所述信息阈值ZI用于表示所有信息子信道中可靠度最低子信道对应的可靠度；所述校验阈值ZP用于表示所有校验子信道中，可靠度大于信息阈值ZI且行重最大的子信道集合中可靠度最低的子信道对应的可靠度。
22.根据权利要求21所述的装置，其特征在于，所述处理模块具体用于：
对于所述信息比特，则将该信息比特对应的路径扩展为0和1的两个路径，并分别计算两条路径的度量值；
如果扩展后的路径数小于或等于L，则保留所有路径；
如扩展后路径数大于L，则保留度量值最优的L条路径。
23.根据权利要求21或22所述的装置，其特征在于，所述处理模块具体用于：
所述校验比特有多个，当所述接收设备译到最后一个校验比特之前，则将该最后一个校验比特之前的校验比特对应的路径扩展为0和1的两个路径，并分别计算两条路径的度量值；
如果扩展后的路径数小于或等于L，则保留所有路径；
如扩展后路径数大于L，则保留度量值最小的L条路径；
所述从度量值最优的候选路径开始，对其中的T条候选路径进行校验包括：
当所述接收设备译到最后一个校验比特时，所述接收设备从保留的l条路径中度量值最优的候选路径开始，使用所述最后一个校验比特对其中的T条候选路径进行校验。
24.根据权利要求21或22所述的装置，其特征在于，所述待译码的比特序列中还包括校验冻结比特，所述校验冻结比特的数量P为如下多个值中的一个：
P＝1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16。
25.一种通信装置，其特征在于，包括处理器和存储器，所述存储器用于存储计算机执行指令，所述处理器用于执行所述计算机执行指令，使得权利要求1-8中任意一项或权利要求17-20中任意一项所述的方法被执行。
26.一种计算机可读存储介质，其特征在于，所述存储介质中存储有指令，当计算机执行所述指令时，权利要求1-8中任意一项或权利要求17-20中任意一项所述的方法被执行。
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