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.

翻译：分组长度为$n$、码本大小为$M$的二进制码称为$(n,M)$码，本文研究其在最大似然（ML）译码下用于无记忆二进制对称信道（BSC）的性能。对于任意$n \geq 2$，先前研究已显式刻画了线性$(n,4)$码中的部分最优码，但线性码中的最优码是否优于所有非线性码尚不清楚。本文首先证明：对于任意$n\geq 2$，存在一个最优码（在所有$(n,4)$码中）要么是线性码，要么属于一类称为I类码的非线性码子集。我们确定了每个分组长度$n\geq 2$下所有线性$(n,4)$码中的最优码，并发现了文献中未给出的码型。对于$n=2$至$300$的所有长度，我们识别了所有最优$(n,4)$码，其中除$n=3$外，所有最优$(n,4)$码均与线性码等价。存在与线性码不等价的最优$(3,4)$码。此外，我们推导了一个称为II类码的非线性码子集，并证明：对于任意$n >300$，由线性码、I类码、II类码及其等价码构成的集合包含了所有最优$(n,4)$码。I类和II类码均接近线性码，其仅涉及一类线性码中未包含的列。我们的结果基于一种比较两码ML译码性能的新技术，该技术通过对信道输出的整个取值范围进行划分实现。