Divisible Codes

TL;DR

This paper discusses $ ext{divisible codes}$, a class of linear codes with all codeword weights divisible by a fixed integer, exploring their properties, applications, and the challenge of determining possible code lengths.

Contribution

The paper reviews the concept of divisible codes, their applications, and highlights the challenge of determining feasible lengths for projective divisible codes.

Findings

Divisible codes have applications in subspace codes and partitions.

Determining possible lengths of projective divisible codes is complex.

The study emphasizes the importance of divisibility in code design.

Abstract

A linear code over $\mathbb{F}_q$ with the Hamming metric is called $\Delta$-divisible if the weights of all codewords are divisible by $\Delta$. They have been introduced by Harold Ward a few decades ago. Applications include subspace codes, partial spreads, vector space partitions, and distance optimal codes. The determination of the possible lengths of projective divisible codes is an interesting and comprehensive challenge.