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IEEE Transactions on Information Theory

Quantum Codes Using the τ\tauτ -OD MP Construction

Meng Cao, Kun Zhou

Published Nov 17, 2025• 4 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2025.3633795

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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.
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.
Asymmetric quantum codes are useful for quantum channels in which phase and bit errors occur with different probabilities, since the two distances, dz and dx, can be controlled separately. We develop a permutation-paired matrix-product construction for such codes over Fq. The main task is to build classical code pairs C,D⊆Fq2kn satisfying the Hermitian inclusion D⊥H⊆C, while keeping explicit dimension and distance bounds. Let A∈Fq2k×k be a non-singular-by-columns (NSC) matrix with AA†=DPτ, where D is an invertible diagonal and Pτ corresponds to an involution τ. For C=[C1,…,Ck]A and D=[D1,…,Dk]A, we prove D⊥H=[Dτ(1)⊥H,…,Dτ(k)⊥H]A. Thus, the global inclusion D⊥H⊆C is equivalent to the shorter paired inclusions Dτ(i)⊥H⊆Ci. This yields asymmetric quantum codes with parameters [[kn,∑i=1k(ri+si)−kn,dz/dx]]q, where the bounds for dz and dx follow from NSC matrix-product distance estimates. For nested maximum distance separable (MDS) constituents, the paired conditions reduce to ri+sτ(i)≥n, giving explicit infinite families. Concrete τ-OD matrices and numerical examples show that nontrivial permutations can increase the quantum dimension while preserving prescribed lower bounds for dz and dx.
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.
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. 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.