On Galois duality, self-orthogonality, and dual-containment of matrix product codes
Ramy Farouk Taki Eldin
TL;DR
This work derives a unified, general framework for Galois duality of matrix product (MP) codes without restricting the defining matrix $\mathcal{A}$ to be square or full rank. It provides a complete set of necessary and sufficient conditions for MP codes to be $\ell$-Galois self-orthogonal and to be $\ell$-Galois dual-containing, treating separately the full-row-rank and rank-deficient cases, and shows how Euclidean and Hermitian duals arise as special cases. The authors present explicit duality formulas, partitions of $\mathcal{A}$ into subcodes when $\mathcal{A}$ is not full rank, and numerous numerical examples that achieve best-known parameters, highlighting potential gains for quantum stabilizer code constructions. The results broaden prior work by removing restrictive assumptions on $\mathcal{A}$ and on $\mathcal{A}\mathcal{A}^T$, and by establishing necessary and sufficient conditions that can guide practical MP-code design in the Galois setting.
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.