A new method for constructing linear codes with small hulls

TL;DR

This paper introduces a novel method for constructing linear codes with small hulls, including LCD and one-dimensional hull codes, using Gauss sums, leading to optimal or near-optimal codes.

Contribution

The paper develops a new construction technique for linear codes with small hulls employing positive characteristic Gauss sums, expanding the class of quasi-abelian and circulant codes.

Findings

Constructed codes include LCD and one-dimensional hull codes.

Provided lower bounds on minimum distances of these codes.

Achieved some optimal or near-optimal codes according to Grassl's database.

Abstract

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull ( LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.