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arXiv:2606.14253

Quantum codes and optimal pure quantum $(r,δ)$-LRCs via the MP construction

Meng Cao, Kun Zhou

quant-phSubmitted Jun 12, 2026• Announced Jun 15, 2026• 0 citations
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Abstract

In this paper, we employ MP codes whose defining matrices are $τ$-optimal defining ($τ$-OD) matrices to construct new quantum codes and quantum $(r,δ)$-LRCs. Specifically, we report the following results: We establish a unified $τ$-monomial decomposition theorem for invertible self-adjoint matrices over finite fields of arbitrary characteristic, which generalizes the result in "Quantum codes using the $τ$-OD MP construction" where the characteristic was required to be odd. Based on this theorem, we prove the existence of $τ$-OD matrices over $\mathbb{F}_{q^2}$ for any characteristic and demonstrate that there exist several new infinite families of $τ$-OD matrices over $\mathbb{F}_{q^2}$ of characteristic $2$. As an application of MP codes involving $τ$-OD matrices, we construct several infinite families of quantum codes with flexible parameters. Within this framework, we present $222$ 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 $(r,δ)$-LRCs via MP codes. Accordingly, we construct four new infinite families of optimal pure quantum $(r,δ)$-LRCs with flexible parameters. Notably, we report an interesting phenomenon by exhibiting $30$ optimal pure quantum $(r,δ)$-LRCs derived from our framework; that is, there exist quantum codes that are not only optimal pure quantum $(r,δ)$-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 $(r,δ)$-LRCs and record-breaking quantum codes has not been previously reported in the literature.

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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 (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.
Optimal pure quantum (r,δ)(r,δ)(r,δ)-locally recoverable codes from matrix-product construction
Locally recoverable codes (LRCs) are classical error-correcting codes widely used in large-scale distributed and cloud storage systems. Quantum locally recoverable codes of locality (r,δ)(r,δ)(r,δ) (quantum (r,δ)(r,δ)(r,δ)-LRCs) are the quantum counterpart of classical (r,δ)(r,δ)(r,δ)-LRCs. They allow us to correct erasures at several positions using a trace-preserving quantum operation acting on qudits of a larger set of positions. Quantum (r,δ)(r,δ)(r,δ)-LRCs, Q(C)\mathcal{Q}(\mathcal{C})Q(C), can be constructed from classical Euclidean (or Hermitian) dual-containing codes C\mathcal{C}C, and their recovery abilities are upper bounded by the minimum distance of the Euclidean (or Hermitian) dual of those codes. Parameters and localities of pure quantum (r,δ)(r,δ)(r,δ)-LRCs satisfy a Singleton-like bound; codes attaining equality are referred to as optimal. We consider matrix-product codes (MPCs) C\mathcal{C}C and give constituent (or defining) matrices and conditions on the constituent codes such that the codes C\mathcal{C}C satisfy the conditions to provide quantum (r,δ)(r,δ)(r,δ)-LRCs. As a consequence, we are able to determine their locality and parameters. Furthermore, we determine families of optimal pure quantum (r,δ)(r,δ)(r,δ)-LRCs derived from them.
In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo et al.et~al.et al. (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for t∈{2,3}t\in \{2,3\}t∈{2,3}, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.
In 2019, Carletet al. introduced the concept of σ duals of linear codes involving the σ inner product, which generalizes the Euclidean, Hermitian and ℓ-Galois cases. This paper focuses on constructing new and improved classical codes and quantum codes within the framework of the σ inner product. We derive some general properties of linear codes, including matrix-product (MP) codes, with respect to the σ inner product. We develop general methods and design effective routes involving certain optimization problems for constructing σ self-orthogonal (SO) and σ dual-containing (DC) MP codes. Our schemes efficiently generate numerous such codes with new or optimal parameters. We establish the σ construction of quantum stabilizer codes from classical codes. We propose a unified method for constructing two general classes of entanglement-assisted quantum error-correcting codes (EAQECCs) based on the σ hulls of general linear codes. This further yields six types of EAQECCs with flexible parameters based on propagation rules using MP codes under the Euclidean and Hermitian cases. Compared to the best-known ternary EAQECCs, we obtain 17 new ones and 13 of them have improved parameters. Finally, we present two infinite families of q-ary EAQECCs with lengths (q2− 1)(q+ 2) andq2(q+2), respectively. These families include many q-ary QECCs that are not only new according to Grassl’s online database but also surpass those listed in Edel’s online database.
Optimal Pure Quantum (r,δ)(r,δ)(r,δ)-LRCs from Euclidean and Hermitian Dual-Containing Cyclic Codes
Locally recoverable codes (LRCs) combine global error correction with the ability to repair a small number of erased coordinates by accessing only a limited number of other coordinates. Motivated by their quantum counterparts, we construct several families of optimal cyclic (r,δ)(r,δ)(r,δ)-LRCs that are Euclidean or Hermitian dual-containing. In the Euclidean case, we obtain four families of optimal dual-containing cyclic (r,δ)(r,δ)(r,δ)-LRCs over Fq\mathbb{F}_qFq​ with a range of minimum distances extending beyond the local distance δδδ. In the Hermitian case, we derive analogous families over Fq2\mathbb{F}_{q^2}Fq2​ that are Hermitian dual-containing, when (r+δ−1)∣(q2−1)(r+δ-1)\mid(q^2-1)(r+δ−1)∣(q2−1). In addition, we develop a distinct construction for the case (r+δ−1)∣(q2+1)(r+δ-1)\mid(q^2+1)(r+δ−1)∣(q2+1) using symmetric defining sets and odd δδδ, which yields optimal codes with minimum distances ℓ+2\ell+2ℓ+2, δ+2δ+2δ+2, 2δ−22δ-22δ−2, and 2δ2δ2δ. The Euclidean Calderbank-Shor-Steane (CSS) and Hermitian stabilizer constructions then give corresponding quantum cyclic (r,δ)(r,δ)(r,δ)-LRCs over~Fq\mathbb{F}_qFq​. Further, the resulting stabilizer codes are pure; they meet the relevant quantum Singleton-type bound, and are therefore \textit{optimal}. We also provide a comparison with previous works, highlighting the parameter regimes and minimum distance ranges covered by our results that are not attained by existing constructions. Explicit examples illustrate the constructions and verify the dual-containment and purity conditions.