Special matrices over finite fields and their applications to quantum error-correcting codes

TL;DR

This paper investigates matrix-product codes over finite fields with special matrix conditions, providing formulas and criteria for their Hermitian dual properties, and constructing quantum error-correcting codes with desirable features.

Contribution

It introduces explicit formulas and conditions for Hermitian dual properties of MP codes with special matrices, advancing quantum code construction methods.

Findings

Explicit formula for Hermitian hull dimension

Necessary and sufficient conditions for dual-containing properties

Construction methods for quantum error-correcting codes

Abstract

The matrix-product (MP) code $\mathcal{C}_{A,k}:=[\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}]\cdot A$ with a non-singular by column (NSC) matrix $A$ plays an important role in constructing good quantum error-correcting codes. In this paper, we study the MP code when the defining matrix $A$ satisfies the condition that $AA^{\dag}$ is $(D,\tau)$-monomial. We give an explicit formula for calculating the dimension of the Hermitian hull of a MP code. We provide the necessary and sufficient conditions that a MP code is Hermitian dual-containing (HDC), almost Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD, respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code with these properties, respectively. We give alternative necessary and sufficient conditions for a MP code to be AHDC and AHSO, respectively, and show several cases where a MP code is not AHDC or AHSO. We provide the construction methods of HDC and AHDC MP codes, including those with optimal minimum distance lower bounds.