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

Meng Cao

Published Dec 31, 2020• 1 citation
OpenAlex

Abstract

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.

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

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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.

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Recently, the construction of matrix-product codes with certain self-orthogonality is an effective way to obtain good quantum codes. One of the most frequently used means in this process is to construct quasi-unitary matrices as the defining matrices of matrix-product codes. Further, if the defining matrix is NSC, then the corresponding matrix-product code has maximal minimum distance. Inspired by these idea, we propose a constructive method about NSC quasi-unitary matrices over Fq2 with q being a prime power, and construct k × k NSC quasi-unitary matrices for any kq[x1, . . . , xk], extending or improving the known results. Through such defining matrices, we obtain new classes of quantum codes from matrix-product codes, many of which show better parameters than previously known.

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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.

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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
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.
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.
Recently, the construction of matrix-product codes with certain self-orthogonality is an effective way to obtain good quantum codes. One of the most frequently used means in this process is to construct quasi-unitary matrices as the defining matrices of matrix-product codes. Further, if the defining matrix is NSC, then the corresponding matrix-product code has maximal minimum distance. Inspired by these idea, we propose a constructive method about NSC quasi-unitary matrices over Fq2 with q being a prime power, and construct k × k NSC quasi-unitary matrices for any kq[x1, . . . , xk], extending or improving the known results. Through such defining matrices, we obtain new classes of quantum codes from matrix-product codes, many of which show better parameters than previously known.
Quasi-cyclic codes have been recently employed in the constructions of quantum error-correcting codes. In this paper, we propose a construction of infinite families of quasi-cyclic codes over $\F_q$ which are self-orthogonal with respect to the Euclidean and Hermitian inner products. In particular, their dimension and a lower bound for their minimum distance are computed using their constituent codes defined over field extensions of Fq\mathbb{F}_qFq​. We also show that the lower bound for the minimum distance satisfies the square-root-like lower bound and also show how dual-containing and self-dual quasi-cyclic codes can arise from our construction. Using the CSS construction, we show the existence of quantum error-correcting codes with good parameters.
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.
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.
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.