An improved method for constructing linear codes with small hulls

TL;DR

This paper introduces a generalized method for constructing linear codes with small hulls, leading to the creation of many optimal Euclidean and Hermitian LCD codes that surpass previous bounds, along with tables of self-dual codes.

Contribution

The paper generalizes existing methods to construct linear codes with small hulls, resulting in numerous optimal LCD codes and improved bounds on minimum distances.

Findings

Many optimal Euclidean LCD codes constructed

Many optimal Hermitian LCD codes constructed

Improved lower bounds on the largest minimum distance

Abstract

In this paper, we give a method for constructing linear codes with small hulls by generalizing the method in \cite{LCD-T-matric}. As a result, we obtain many optimal Euclidean LCD codes and Hermitian LCD codes, which improve the previously known lower bound on the largest minimum distance. We also obtain many optimal codes with one-dimension hull. Furthermore, we give three tables about formally self-dual LCD codes.