On Optimal Finite-length Block Codes of Size Four for Binary Symmetric Channels

An (
n
,
M
) code refers to a binary code with blocklength
n
and codebook size
M
. Such codes are studied in the context of memoryless binary symmetric channels (BSCs) with maximum likelihood (ML) decoding. Previous research has characterized some optimal codes among the linear (
n
, 4) codes for any
n
≥ 2. However, it was unknown whether these optimal codes among linear codes were better than all nonlinear codes. In this paper, we first demonstrate that for any
n
≥ 2, there exists an optimal code among all (
n
, 4) codes that is either linear or belongs to a subset of nonlinear codes called Class-I codes. We identify all the optimal codes among the linear (
n
, 4) codes for each blocklength
n
≥ 2 and discover some that were not previously reported in the literature. For any
n
from 2 to 8, all the optimal (
n
, 4) codes are identified. 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 introduce a subset of nonlinear codes called Class-II codes and show that for any
n
> 3, 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 column that is not included in linear codes. We derive a sufficient condition such that all the optimal (
n
, 4) codes are equivalent to linear codes, which can be evaluated by computer with a computation cost
O
(
n
⁶
).