On the binary linear constant weight codes and their autormorphism groups

TL;DR

This paper characterizes binary linear constant weight codes using support symmetric differences, establishes a formula for their automorphism groups, and provides an alternative proof of their equivalence, advancing algebraic understanding of these codes.

Contribution

It introduces a novel characterization of binary linear constant weight codes via support symmetric differences and derives a formula for their automorphism group order.

Findings

A new support-based characterization of the codes.

A formula for automorphism group order in terms of code parameters.

An alternative proof of code equivalence using explicit permutations.

Abstract

We give a characterization for the binary linear constant weight codes by using the symmetric difference of the supports of the codewords. This characterization gives a correspondence between the set of binary linear constant weight codes and the set of partitions for the union of supports of the codewords. By using this correspondence, we present a formula for the order of the automorphism group of a binary linear constant weight code in terms of its parameters. This formula is a key step to determine more algebraic structures on constant weight codes with given parameters. Bonisoli [Bonisoli, A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Combinatoria 18, 181--186 (1984)] proves that the $q$-ary linear constant weight codes with the same parameters are equivalent (for the binary case permutation equivalent). We also give an alternative proof for Bonisoli's theorem by presenting an explicit permutation on symmetric difference of the supports of the codewords which gives the permutation equivalence between the binary linear constant weight codes.