On the Hulls of Group Codes

TL;DR

This paper establishes a general criterion for the existence of group codes with specified hull dimensions in group algebras over finite fields, extending previous results to non-abelian groups and small hull dimensions.

Contribution

It introduces a new criterion for the existence of group codes with given hull dimensions, applicable to non-abelian groups and small hull sizes, generalizing prior abelian-only results.

Findings

Criterion for existence of group codes with given hull dimension

Explicit conditions for hull dimension ≤ 3

Generalization to non-abelian groups

Abstract

Let $\mathbb {F}_q$ be a finite field and $G$ a finte group with $(|G|,q)=1$. By a group code in $\mathbb {F}_q[G]$ we mean a two-sided ideal in $\mathbb {F}_q[G]$. We will prove a general criterion for the existence of group codes with given hull dimension, and then apply it to deduce explicit criterions for existence of group codes with hull dimension $\leq3$. In particular our criterion for the existence of $1$-dimensional hulls generalizes that of privious work which consider only abelian groups $G$.