On Abelian one-dimensional hull codes in group algebras

Abstract

This paper focuses on hull dimensional codes obtained by the intersection of linear codes and their dual. These codes were introduced by Assmus and Key and have been the subject of significant theoretical and practical research over the years, gaining increased attention in recent years. Let \(\mathbb {F}_q\) denote the finite field with q elements, and let G be a finite Abelian group of order n. The paper investigates Abelian codes defined as ideals of the group algebra \(\mathbb {F}_qG\) with coefficients in \(\mathbb {F}_q\). Specifically, it delves into Abelian hull dimensional codes in the group algebra \(\mathbb {F}_qG\), where G is a finite Abelian group of order n with \(\gcd (n,q)=1\). Specifically, we first examine general hull Abelian codes and then narrow its focus to Abelian one-dimensional hull codes. Next, we focus on Abelian one-dimensional hull codes and present some necessary and sufficient conditions for characterizing them. Consequently, we generalize a recent result on Abelian codes and show that no binary or ternary Abelian codes with one-dimensional hulls exist. Furthermore, we construct Abelian codes with one-dimensional hulls by generating idempotents, derive optimal ones with one-dimensional hulls, and establish several existing results of Abelian codes with one-dimensional hulls. Finally, we develop enumeration results through a simple formula that counts Abelian codes with one-dimensional hulls in \(\mathbb {F}_qG\). These achievements exploit the rich algebraic structure of those Abelian codes and enhance and increase our knowledge of them by considering their hull dimensions, reducing the gap between their interests and our understanding of them.