Optimal Binary Constant Weight Codes and Affine Linear Groups over Finite Fields

TL;DR

This paper investigates the structure of binary constant weight codes derived from affine linear groups over finite fields, providing a sum formula that characterizes possible code parameters and determining the maximum size of such codes meeting the Johnson bound.

Contribution

It introduces a sum formula that characterizes stabilizers in affine linear groups and determines all parameters of certain binary constant weight codes meeting the Johnson bound.

Findings

Derived a sum formula for stabilizers in affine linear groups.

Determined all possible parameters of binary constant weight codes from group actions.

Calculated the maximum size of codes for many parameters, matching the Johnson bound.

Abstract

Let $\text{AGL}(1,\Bbb F_q)$ be the affine linear group of dimension $1$ over a finite field $\Bbb F_q$. $\text{AGL}(1,\Bbb F_q)$ acts sharply 2-transitively on $\Bbb F_q$. Given $S<\text{AGL}(1,\Bbb F_q)$ and an integer $k$ with $1\le k\le q$, does there exist a subset $B\subset\Bbb F_q$ with $|B|=k$ such that $S=\text{AGL}(1,\Bbb F_q)_B$? ($\text{AGL}(1,\Bbb F_q)_B=\{\sigma\in\text{AGL}(1,\Bbb F_q):\sigma(B)=B\}$ is the stabilizer of $B$ in $\text{AGL}(1,\Bbb F_q)$.) We derive a sum that holds the answer to this question. This result determines all possible parameters of binary constant weight codes that are constructed from the action of $\text{AGL}(1,\Bbb F_q)$ on $\Bbb F_q$ to meet the Johnson bound. Consequently, the values of the function $A_2(n,d,w)$ are determined for many parameters, where $A_2(n,d,w)$ is the maximum number of codewords in a binary constant weight code of length $n$, weight $w$ and minimum distance $\ge d$.