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Mathematics

Asymmetric Quantum Codes from τ-Paired Matrix-Product Codes

Sami Alabiad, Shayea Aldossari

Published Jun 21, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.3390/math14122226

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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.
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.
Recently, the theory of quantum error control codes has been extended to include quantum codes over asymmetric quantum channels - qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper we establish a method to construct asymmetric quantum codes based on classical codes. We derive, once again, families of asymmetric quantum codes from classical BCH and RS codes over finite fields. Particularly, we present interesting asymmetric quantum codes based on nonprimitive narrow-sense BCH codes with parameters [[n, k, dz/dx]]qfor certain values of code lengths, dimensions, and various minimum distance. Finally, our constructions are well explained by an illustrative example.
The theory of quantum error correcting codes is a primary tool for fighting decoherence and other quantum noise in quantum communication and quantum computation. Recently, the theory of quantum error correcting codes has developed rapidly and been extended to protect quantum information over asymmetric quantum channels, in which phase-shift and qubit-flip errors occur with different probabilities. In this paper, we generalize the construction of symmetric quantum codes via graphs (or matrices) to the asymmetric case, converting the construction of asymmetric quantum codes to finding matrices with some special properties. We also propose some asymmetric quantum Maximal Distance Separable (MDS) codes as examples constructed in this way.
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.
We construct three new families of asymmetric quantum MDS codes from nested Hermitian self-orthogonal generalized Reed–Solomon and extended generalized Reed–Solomon codes over Fq2. The construction is developed in three settings: affine partitions of Fq2, projective norm partitions of Fq2*, and extended affine configurations obtained by adjoining the point at infinity. In each case, the Hermitian orthogonality conditions are reduced to explicit linear systems over Fq, whose solvability follows from structured moment identities and Vandermonde-type arguments. This yields nested classical MDS codes satisfying the Hermitian dual-containment condition required in the Hermitian construction of asymmetric quantum codes. As a consequence, we obtain three explicit families of asymmetric quantum MDS codes with fully determined lengths, dimensions, and asymmetric distances dz and dx. Our results show that affine and projective partition techniques provide a natural and effective framework for constructing optimal asymmetric quantum codes with flexible parameters.