On Galois duality, self-orthogonality, and dual-containment of matrix product codes

TL;DR

This paper provides a comprehensive formula for Galois duals of matrix product codes, establishing new necessary and sufficient conditions for self-orthogonality and dual-containment applicable to any defining matrix, broadening previous results.

Contribution

It introduces a general characterization of Galois duals for MP codes with arbitrary defining matrices, extending beyond invertible or structured matrices and broadening duality analysis.

Findings

Derived a formula for Galois duals of MP codes with any defining matrix.

Established necessary and sufficient conditions for self-orthogonality and dual-containment.

Provided numerical examples demonstrating the theoretical results.

Abstract

In recent literature, matrix product (MP) codes and their duals have gained significant attention due to their application in the construction of quantum stabilizer codes. In this paper, we begin with providing a formula that characterizes the Galois dual of MP codes. Using this formula, we establish the conditions under which MP codes are self-orthogonal and dual-containing. Although similar results may exist in the literature, the novelty and superiority of our results can be identified in the following points. Previous results that characterize the duals of MP codes only apply to MP codes with an invertible square defining matrix $\mathcal{A}$. However, our characterization applies to MP code with any defining matrix, whether $\mathcal{A}$ is not square or not of full row rank. Previous studies on the conditions for self-orthogonality or dual-containment of MP codes have assumed certain structures for the product $\mathcal{A}\mathcal{A}^T$ or $\mathcal{A}\mathcal{A}^{\dagger}$, such as being diagonal, anti-diagonal, monomial, or partitioned Hermitian orthogonal. However, our conditions do not necessitate such specific structures. Previous studies investigated MP code duality in the context of Euclidean and Hermitian duals; however, we investigate MP code duality in the broader context of Galois dual, with Euclidean and Hermitian duals acting as special cases. Finally, it is worth noting that the proposed conditions for Galois self-orthogonality or dual-containment are both necessary and sufficient. To demonstrate the theoretical results, several numerical examples with best-known parameters MP codes are provided.