Construction of optimal Hermitian self-dual codes from unitary matrices

TL;DR

This paper introduces an algorithm for constructing unitary matrices over finite fields and uses them to generate new optimal Hermitian self-dual codes, including many MDS and almost MDS codes with large parameters.

Contribution

The paper presents novel constructions of Hermitian self-dual codes using unitary matrices, extending quadratic double circulant methods and producing codes with new parameters.

Findings

Many optimal Hermitian self-dual codes over large finite fields are constructed.

Constructs MDS or almost MDS Hermitian self-dual codes up to length 18.

Comparisons show advantages over classical constructions.

Abstract

We provide an algorithm to construct unitary matrices over finite fields. We present various constructions of Hermitian self-dual code by means of unitary matrices, where some of them generalize the quadratic double circulant constructions. Many optimal Hermitian self-dual codes over large finite fields with new parameters are obtained. More precisely MDS or almost MDS Hermitian self-dual codes of lengths up to $18$ are constructed over finite fields $\F_{q},$ where $q=3^2,4^2,5^2,7^2,8^2,9^2,11^2,13^2,17^2,19^2.$ Comparisons with classical constructions are made.