Classification of $\Delta$-divisible linear codes spanned by codewords of weight $\Delta$

TL;DR

This paper classifies all q-ary Δ-divisible linear codes spanned by codewords of weight Δ, extending previous binary results and providing new insights into their structure and applications.

Contribution

It generalizes the classification of binary self-orthogonal codes spanned by weight-4 codewords to all q-ary Δ-divisible codes, identifying fundamental building blocks.

Findings

Classified all q-ary Δ-divisible codes spanned by codewords of weight Δ.

Identified simplex codes, Reed-Muller codes, and parity check codes as fundamental components.

Provided an alternative proof of Liu's theorem on binary Δ-divisible codes of length 4Δ.

Abstract

We classify all $q$-ary $\Delta$-divisible linear codes which are spanned by codewords of weight $\Delta$. The basic building blocks are the simplex codes, and for $q=2$ additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight $4$ have been classified, which is the case $q=2$ and $\Delta=4$ of our classification. As an application, we give an alternative proof of a theorem of Liu on binary $\Delta$-divisible codes of length $4\Delta$ in the projective case.