On projective $q^r$-divisible codes

TL;DR

This paper surveys known results on the lengths of projective $q^r$-divisible linear codes over finite fields, which are important in applications like bounds on partial spreads.

Contribution

It provides a comprehensive overview of the current knowledge on the lengths of projective $q^r$-divisible codes, highlighting their significance in finite geometry.

Findings

Summarizes known length bounds for projective $q^r$-divisible codes.

Connects $q^r$-divisible codes to applications in finite geometry.

Identifies open problems and future research directions.

Abstract

A projective linear code over $\mathbb{F}_q$ is called $\Delta$-divisible if all weights of its codewords are divisible by $\Delta$. Especially, $q^r$-divisible projective linear codes, where $r$ is some integer, arise in many applications of collections of subspaces in $\mathbb{F}_q^v$. One example are upper bounds on the cardinality of partial spreads. Here we survey the known results on the possible lengths of projective $q^r$-divisible linear codes.