The $σ$ hulls of matrix-product codes and related entanglement-assisted quantum error-correcting codes

TL;DR

This work addresses constructing entanglement-assisted quantum error-correcting codes (EAQECCs) from matrix-product (MP) codes under a generalized $\\sigma$-inner product. It derives an explicit formula for the dimension of the $\\sigma$-hull of MP codes, establishes dual-containing and self-orthogonality conditions, and proves a symmetry of hull dimensions between a code and its $\\sigma$-dual. The authors then show how to realize any admissible hull dimension $h$ via monomial transforms, enabling EAQECCs with tunable parameters $[[n,k-h,d;n-k-h]]_{q}$ and $[[n,n-k-h,d';k-h]]_{q}$ for all $0\\le h\\le \\dim_{\\mathbb{F}_{q}}(\\mathrm{Hull}_{\\sigma}(C))$, and provide a general constructive framework that yields flexible EAQECCs from MP codes, including MDS cases. Overall, the paper broadens systematic EAQECC construction by leveraging $\\sigma$-hulls and monomial equivalence, extending prior results to wider semilinear isometries and defining matrices.

Abstract

Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. Matrix-product (MP) codes are a class of long classical codes generated by combining several commensurate classical codes with a defining matrix. We give an explicit formula for calculating the dimension of the $σ$ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be $σ$ dual-containing and $σ$ self-orthogonal. We prove that $\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))=\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}^{\bot_σ}))$. We prove that for any integer $h$ with $\mathrm{max}\{0,k_{1}-k_{2}\}\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_σ})$, there exists a linear code $\mathcal{C}_{2,h}$ monomially equivalent to $\mathcal{C}_{2}$ such that $\mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2,h}^{\bot_σ})=h$, where $\mathcal{C}_{i}$ is an $[n,k_{i}]_{q}$ linear code for $i=1,2$. We show that given an $[n,k,d]_{q}$ linear code $\mathcal{C}$, there exists a monomially equivalent $[n,k,d]_{q}$ linear code $\mathcal{C}_{h}$, whose $σ$ dual code has minimum distance $d'$, such that there exist an $[[n,k-h,d;n-k-h]]_{q}$ EAQECC and an $[[n,n-k-h,d';k-h]]_{q}$ EAQECC for every integer $h$ with $0\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))$. Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to $σ$ hulls.

Theorems & Definitions (25)

• Remark 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 3.1
• proof
• Theorem 3.2
• proof