Quantum error-correcting codes from matrix-product codes related to quasi-orthogonal and quasi-unitary matrices

TL;DR

This paper explores the construction of quantum error-correcting codes using matrix-product codes over finite fields, focusing on quasi-orthogonal and quasi-unitary matrices to improve code properties.

Contribution

It summarizes previous results and introduces new findings on quasi-orthogonal and quasi-unitary matrices for quantum code construction.

Findings

New classes of quantum codes with improved parameters

Enhanced understanding of dual-containing matrix-product codes

Extension of code construction methods to larger finite fields

Abstract

Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good $q$-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and $q$-ary quantum codes derived from these matrix-product codes.