On the Hulls of Group Codes

Let ${\mathbb{F}}_q$ be a finite field and $G$ a finte group with $(\mathrm{\mid }G\mathrm{\mid },q)=1$. 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 $\le 3$. In particular our criterion for the existence of $1$-dimensional hulls generalizes that of privious work which consider only abelian groups $G$.