论文标题
关于二元对称通道的最佳有限长度块四大代码
On Optimal Finite-length Block Codes of Size Four for Binary Symmetric Channels
论文作者
论文摘要
一个blocklength $ n $的二进制代码和代码书尺寸$ m $称为$（n，m）$代码，该代码用于无内存的二进制对称频道（BSC），并具有最大可能性（ML）解码。对于任何$ n \ geq 2 $，线性$（n，4）$代码之间的某些最佳代码在上一项研究中已明确表征，但是线性代码之间的最佳代码是否比所有非线性代码更好。在本文中，我们首先证明，对于任何$ n \ geq 2 $，都存在一个最佳代码（在所有$（n，4）$代码中，它是线性的，或在非线性代码的子集中，称为类-I类代码。我们确定了每个模块长度$ n \ geq 2 $的线性$（n，4）$代码之间的所有最佳代码，并找到了文献中未给出的。对于任何$ n $从$ 2 $到$ 300 $，所有最佳$（n，4）$代码都被标识，除了$ n = 3 $，所有最佳$（n，4）$代码等于线性代码。存在最佳$（3,4）$代码，这些代码不等于线性代码。此外，我们得出了一个名为II类代码的非线性代码的子集，并证明对于任何$ n> 300 $，该集合由线性，I类和II类代码组成，其等效代码包含所有最佳$（n，4）$代码。 I级和II类代码都靠近线性代码，因为它们仅涉及一种未包含在线性代码中的类型列。我们的结果是使用新技术获得的，以比较两个代码的ML解码性能，并以通道输出的整个范围的分区为特色。
A binary code of blocklength $n$ and codebook size $M$ is called an $(n,M)$ code, which is studied for memoryless binary symmetric channels (BSCs) with the maximum likelihood (ML) decoding. For any $n \geq 2$, some optimal codes among the linear $(n,4)$ codes have been explicitly characterized in the previous study, but whether the optimal codes among the linear codes are better than all the nonlinear codes or not is unknown. In this paper, we first show that for any $n\geq 2$, there exists an optimal code (among all the $(n,4)$ codes) that is either linear or in a subset of nonlinear codes, called Class-I codes. We identified all the optimal codes among the linear $(n,4)$ codes for each blocklength $n\geq 2$, and found ones that were not given in literature. For any $n$ from $2$ to $300$, all the optimal $(n,4)$ codes are identified, where except for $n=3$, all the optimal $(n,4)$ codes are equivalent to linear codes. There exist optimal $(3,4)$ codes that are not equivalent to linear codes. Furthermore, we derive a subset of nonlinear codes called Class-II codes and justify that for any $n >300$, the set composed of linear, Class-I and Class-II codes and their equivalent codes contains all the optimal $(n,4)$ codes. Both Class-I and Class-II codes are close to linear codes in the sense that they involve only one type of columns that are not included in linear codes. Our results are obtained using a new technique to compare the ML decoding performance of two codes, featured by a partition of the entire range of the channel output.
