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IEEE Transactions on Information Theory

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

Meng Cao, Kun Zhou

Published Apr 7, 2026• 1 citation
Publisher

Abstract

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.

DOI
10.1109/tit.2026.3681560

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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.
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.
Quantum codes and optimal pure quantum (r,δ)(r,δ)(r,δ)-LRCs via the MP construction
In this paper, we employ MP codes whose defining matrices are τττ-optimal defining (τττ-OD) matrices to construct new quantum codes and quantum (r,δ)(r,δ)(r,δ)-LRCs. Specifically, we report the following results: We establish a unified τττ-monomial decomposition theorem for invertible self-adjoint matrices over finite fields of arbitrary characteristic, which generalizes the result in "Quantum codes using the τττ-OD MP construction" where the characteristic was required to be odd. Based on this theorem, we prove the existence of τττ-OD matrices over Fq2\mathbb{F}_{q^2}Fq2​ for any characteristic and demonstrate that there exist several new infinite families of τττ-OD matrices over Fq2\mathbb{F}_{q^2}Fq2​ of characteristic 222. As an application of MP codes involving τττ-OD matrices, we construct several infinite families of quantum codes with flexible parameters. Within this framework, we present 222222222 record-breaking quantum codes that surpass the best-known records maintained in Grassl's database. We propose two effective schemes for constructing optimal pure quantum (r,δ)(r,δ)(r,δ)-LRCs via MP codes. Accordingly, we construct four new infinite families of optimal pure quantum (r,δ)(r,δ)(r,δ)-LRCs with flexible parameters. Notably, we report an interesting phenomenon by exhibiting 303030 optimal pure quantum (r,δ)(r,δ)(r,δ)-LRCs derived from our framework; that is, there exist quantum codes that are not only optimal pure quantum (r,δ)(r,δ)(r,δ)-LRCs but also, according to Grassl's database, best-known, optimal, or record-breaking quantum codes. To the best of our knowledge, the new discovery that quantum codes are simultaneously optimal pure quantum (r,δ)(r,δ)(r,δ)-LRCs and record-breaking quantum codes has not been previously reported in the literature.
In this paper, we propose an effective method called the τ-optimal defining ( τ-OD) matrix-product (MP) construction to derive infinite families of quantum codes with flexible parameters. Through this scheme, we present 100 record-breaking quantum codes, which exceed the best-known lower bounds on the minimum distances of quantum codes listed in Grassl’s online database. To this end, we need to construct infinite families of τ-OD matrices, a subclass of non-singular by columns (NSC) matrices. We begin by presenting two key results concerning matrices over general fields F with characteristic char(F) ≠ 2, which involve the construction of self-adjoint τ-monomial matrices from any invertible self-adjoint matrix. The finite field version of these results not only reveals the existence of τ-OD matrices over Fq2for some permutation τ but also provides a general method for constructing τ-OD matrices over Fq2for any permutation τ. We show that there exist several infinite families of τ-OD matrices over Fq2where the permutations τ can be easily expressed. By utilizing these constructed τ-OD matrices, we present several infinite families of quantum codes with flexible parameters, many of which are record-breaking.
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.
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 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.