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arXiv:2405.02285

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

Cao, Meng

Published May 3, 2024• 0 citations
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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 $AA^†$ is $(D,τ)$-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.

DOI
10.48550/arxiv.2405.02285

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

Cao, Meng

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.

arXiv:2405.07740•May 13, 2024•0 citations

Quantum error-correcting codes from matrix-product codes related to quasi-orthogonal matrices and quasi-unitary matrices.

Meng Cao

Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good $q$-ary quantum codes of large length. Specifically, it follows from CSS construction (resp. Hermitian construction) that a matrix-product code over $\mathbb{F}_{q}$ (resp. $\mathbb{F}_{q^{2}}$) which is Euclidean dual-containing (resp. Hermitian dual-containing) can produce a $q$-ary quantum code. In order to obtain such matrix-product codes, a common way is to construct quasi-orthogonal matrices (resp. quasi-unitary matrices) as the defining matrices of matrix-product codes over $\mathbb{F}_{q}$ (resp. $\mathbb{F}_{q^{2}}$). The usage of NSC quasi-orthogonal matrices or NSC quasi-unitary matrices in this process enables the minimum distance lower bound of the corresponding quantum codes to reach its optimum. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in \cite{Cao2020Constructioncaowang,Cao2020New,Cao2020Constructionof}; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and $q$-ary quantum codes derived from these newly constructed matrix-product codes.

•Dec 31, 2020•1 citation

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

Ramy Taki Eldin

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.

arXiv:2408.02799•Aug 5, 2024•0 citations

Construction of New Matrix-Product Codes and Their Applications

Hao Song, Ruihu Li, Junli Wang +1 more

In this paper, from special code chains$\mathcal {C}_{1} \supseteq \mathcal {C}_{2}~\supseteq \mathcal {C}_{3}$such that$\mathcal {C}^{\bot _{h}}_{1}\subseteq \mathcal {C}_{3}$and$\mathcal {C}^{\bot _{h}}_{2}\subseteq \mathcal {C}_{2}$, some Hermitian dual-containing (HDC) matrix-product (MP) codes are presented, where$\mathcal {C}_{3}$is not HDC. By studying some constacyclic codes of lengths$n=q^{2}\pm 1$and$n=\frac {q^{2}-1}{2}$, we construct many HDC MP codes of length$3n$. Consequently, many$q$-ary quantum codes with larger designed distance$d\geq q+1$are obtained from these MP codes, where$4\leq q\leq 9$.

IEEE Access•Jan 1, 2019•3 citations

Quantum error-correcting codes from matrix-product codes related to quasi-orthogonal and quasi-unitary matrices

Cao, Meng

Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good $q$-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and $q$-ary quantum codes derived from these matrix-product codes.

arXiv:2012.15691•Dec 31, 2020•0 citations

An Open Problem on NSC Quasi-Unitary Matrices and Its Application to Quantum Codes

Meng Cao, Kun Zhou

Matrix-product (MP) codes provide an efficient approach for constructing long classical codes by combining a defining matrix with several shorter constituent codes. Employing non-singular by columns (NSC) quasi-unitary matrices as the defining matrices of MP codes enables the construction of various types of quantum codes. The existence of infinite families ofN×NNSC quasi-unitary matrices over Fq2forN = qorq+2 ≤ N ≤q2, whereq≠ 2 is a prime power, was posed as an open problem in the literature and remains largely unresolved. In this paper, we introduce a novel method for constructing NSC quasi-unitary matrices based on a generalized orthogonality relation for cyclic groups. We prove that (2m+ 1) × (2m+ 1) NSC quasi-unitary matrices always exist over Fq2whenever 2m| (q2–1). This result establishes, for the first time, the existence of six new infinite families of NSC quasi-unitary matrices, providing solutions to the aforementioned open problem. The number of such matrices within our framework is also determined. As a key application, we leverage NSC quasi-unitary matrices to construct optimal pure quantum (r; δ)-locally repairable codes (LRCs). We propose an effective method for constructing optimal pure quantum (r; δ)- LRCs from MP codes whose defining matrices are NSC quasiunitary matrices. Based on this method, we explicitly construct two new infinite families of optimal pure quantum (r; δ)-LRCs with flexible parameters.

IEEE Transactions on Information Theory•Apr 7, 2026•1 citation
The matrix-product (MP) code CA,k:=[C1,C2,…,Ck]⋅A\mathcal{C}_{A,k}:=[\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}]\cdot ACA,k​:=[C1​,C2​,…,Ck​]⋅A with a non-singular by column (NSC) matrix AAA plays an important role in constructing good quantum error-correcting codes. In this paper, we study the MP code when the defining matrix AAA satisfies the condition that AA†AA^†AA† is (D,τ)(D,τ)(D,τ)-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.
The σσσ hulls of matrix-product codes and related entanglement-assisted quantum error-correcting codes
Let SLAut(Fqn)\mathrm{SLAut}(\mathbb{F}_{q}^{n})SLAut(Fqn​) denote the group of all semilinear isometries on Fqn\mathbb{F}_{q}^{n}Fqn​, where q=peq=p^{e}q=pe 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 σσσ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be σσσ dual-containing and σσσ self-orthogonal. We prove that dimFq(Hullσ(C))=dimFq(Hullσ(C⊥σ))\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))=\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}^{\bot_σ}))dimFq​​(Hullσ​(C))=dimFq​​(Hullσ​(C⊥σ​)). We prove that for any integer hhh with max{0,k1−k2}≤h≤dimFq(C1∩C2⊥σ)\mathrm{max}\{0,k_{1}-k_{2}\}\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_σ})max{0,k1​−k2​}≤h≤dimFq​​(C1​∩C2⊥σ​​), there exists a linear code C2,h\mathcal{C}_{2,h}C2,h​ monomially equivalent to C2\mathcal{C}_{2}C2​ such that dimFq(C1∩C2,h⊥σ)=h\mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2,h}^{\bot_σ})=hdimFq​​(C1​∩C2,h⊥σ​​)=h, where Ci\mathcal{C}_{i}Ci​ is an [n,ki]q[n,k_{i}]_{q}[n,ki​]q​ linear code for i=1,2i=1,2i=1,2. We show that given an [n,k,d]q[n,k,d]_{q}[n,k,d]q​ linear code C\mathcal{C}C, there exists a monomially equivalent [n,k,d]q[n,k,d]_{q}[n,k,d]q​ linear code Ch\mathcal{C}_{h}Ch​, whose σσσ dual code has minimum distance d′d'd′, such that there exist an [[n,k−h,d;n−k−h]]q[[n,k-h,d;n-k-h]]_{q}[[n,k−h,d;n−k−h]]q​ EAQECC and an [[n,n−k−h,d′;k−h]]q[[n,n-k-h,d';k-h]]_{q}[[n,n−k−h,d′;k−h]]q​ EAQECC for every integer hhh with 0≤h≤dimFq(Hullσ(C))0\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))0≤h≤dimFq​​(Hullσ​(C)). Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to σσσ hulls.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. Specifically, it follows from CSS construction (resp. Hermitian construction) that a matrix-product code over Fq\mathbb{F}_{q}Fq​ (resp. Fq2\mathbb{F}_{q^{2}}Fq2​) which is Euclidean dual-containing (resp. Hermitian dual-containing) can produce a qqq-ary quantum code. In order to obtain such matrix-product codes, a common way is to construct quasi-orthogonal matrices (resp. quasi-unitary matrices) as the defining matrices of matrix-product codes over Fq\mathbb{F}_{q}Fq​ (resp. Fq2\mathbb{F}_{q^{2}}Fq2​). The usage of NSC quasi-orthogonal matrices or NSC quasi-unitary matrices in this process enables the minimum distance lower bound of the corresponding quantum codes to reach its optimum. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in \cite{Cao2020Constructioncaowang,Cao2020New,Cao2020Constructionof}; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these newly constructed matrix-product codes.
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 A\mathcal{A}A. However, our characterization applies to MP code with any defining matrix, whether A\mathcal{A}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 AAT\mathcal{A}\mathcal{A}^TAAT or AA†\mathcal{A}\mathcal{A}^{\dagger}AA†, 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.
In this paper, from special code chainsC1⊇C2 ⊇C3\mathcal {C}_{1} \supseteq \mathcal {C}_{2}~\supseteq \mathcal {C}_{3}C1​⊇C2​ ⊇C3​such thatC1⊥h⊆C3\mathcal {C}^{\bot _{h}}_{1}\subseteq \mathcal {C}_{3}C1⊥h​​⊆C3​andC2⊥h⊆C2\mathcal {C}^{\bot _{h}}_{2}\subseteq \mathcal {C}_{2}C2⊥h​​⊆C2​, some Hermitian dual-containing (HDC) matrix-product (MP) codes are presented, whereC3\mathcal {C}_{3}C3​is not HDC. By studying some constacyclic codes of lengthsn=q2±1n=q^{2}\pm 1n=q2±1andn=q2−12n=\frac {q^{2}-1}{2}n=2q2−1​, we construct many HDC MP codes of length3n3n3n. Consequently, manyqqq-ary quantum codes with larger designed distanced≥q+1d\geq q+1d≥q+1are obtained from these MP codes, where4≤q≤94\leq q\leq 94≤q≤9.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these matrix-product codes.
Matrix-product (MP) codes provide an efficient approach for constructing long classical codes by combining a defining matrix with several shorter constituent codes. Employing non-singular by columns (NSC) quasi-unitary matrices as the defining matrices of MP codes enables the construction of various types of quantum codes. The existence of infinite families ofN×NNSC quasi-unitary matrices over Fq2forN = qorq+2 ≤ N ≤q2, whereq≠ 2 is a prime power, was posed as an open problem in the literature and remains largely unresolved. In this paper, we introduce a novel method for constructing NSC quasi-unitary matrices based on a generalized orthogonality relation for cyclic groups. We prove that (2m+ 1) × (2m+ 1) NSC quasi-unitary matrices always exist over Fq2whenever 2m| (q2–1). This result establishes, for the first time, the existence of six new infinite families of NSC quasi-unitary matrices, providing solutions to the aforementioned open problem. The number of such matrices within our framework is also determined. As a key application, we leverage NSC quasi-unitary matrices to construct optimal pure quantum (r; δ)-locally repairable codes (LRCs). We propose an effective method for constructing optimal pure quantum (r; δ)- LRCs from MP codes whose defining matrices are NSC quasiunitary matrices. Based on this method, we explicitly construct two new infinite families of optimal pure quantum (r; δ)-LRCs with flexible parameters.