关于矩阵乘积码的伽罗瓦对偶、自正交性与对偶包含性
On Galois duality, self-orthogonality, and dual-containment of matrix product codes
AI总结：
本文给出了任意定义矩阵下矩阵乘积码的伽罗瓦对偶公式，建立了无需特定矩阵结构且充分必要的自正交与对偶包含条件，推广了欧几里得和埃尔米特对偶情形，并用最佳已知参数示例验证了理论结果。
摘要：
在近期文献中，矩阵乘积（MP）码及其对偶码因其在量子稳定子码构造中的应用而受到广泛关注。本文首先给出了刻画 MP 码的伽罗瓦对偶的公式。利用该公式，我们建立了 MP 码自正交和对偶包含的条件。尽管文献中可能存在类似结果，但我们的结果的新颖性和优越性体现在以下几点。以往刻画 MP 码对偶的结果仅适用于具有可逆方阵定义矩阵 $\mathcal{A}$ 的 MP 码。然而，我们的刻画适用于具有任意定义矩阵的 MP 码，无论 $\mathcal{A}$ 不是方阵还是行满秩不成立。以往关于 MP 码自正交性或对偶包含性条件的研究都假设乘积 $\mathcal{A}\mathcal{A}^T$ 或 $\mathcal{A}\mathcal{A}^{\dagger}$ 具有特定结构，例如对角、反对角、单项式或分块埃尔米特正交。然而，我们的条件不需要这些特定结构。以往研究在欧几里得对偶和埃尔米特对偶的背景下研究了 MP 码对偶性；而我们则在更广泛的伽罗瓦对偶背景下研究 MP 码对偶性，欧几里得对偶和埃尔米特对偶作为特例。最后，值得指出的是，所提出的伽罗瓦自正交性或对偶包含性条件既是必要的也是充分的。为验证理论结果，提供了若干具有最佳已知参数的 MP 码数值示例。
英文摘要：
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.
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