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IEEE Transactions on Information Theory

Classical Codes and Quantum Codes Involving the σ Inner Product

Meng Cao, Yang Li, Shixin Zhu

Published Jul 31, 2025• 3 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2025.3594472

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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.
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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.
The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding classical self-orthogonal linear codes over the Galois field GF(4) under a Hermitian inner product.A classical cyclic code over the field GF(4) is self-orthogonal with respect to the Hermitian inner product iff the generator polynomial of its dual code is a factor of its generator polynomial.Based on this connection,a class of quantum cyclic codes is constructed by finding corresponding classical cyclic codes.Some examples are also given in detail.
It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 3n-length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.
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.
The classical linear codes that construct quantum error correction codes need to meet certain specific conditions. The proposal of entanglement-assisted quantum error correcting codes eliminates the restriction of these conditions, so all classical linear codes can be constructed into entanglement-assisted quantum error correcting codes. The Hadamard matrix is a special type of orthogonal matrix. Hadamard codes, constructed based on the standard form of Hadamard matrices, are a type of linear complementary dual (LCD) codes. By extending the generator and parity-check matrices of Hadamard codes, two types of classical codes whose parameters are [(l+1)(n−1), n−1] and [3n−3, n−1] maintain the LCD property. Utilizing the relationship between their LCD property and the dimensions of the Euclidean hull, this paper constructs two types of EAQECCs based on Hadamard matrices. These EAQECCs can achieve maximal entanglement, resulting in higher code rates and net rates compared to other EAQECCs. Their parameters are [[3n−3, 2n−2, d≥1; n−1]] and [[2n−2, n−1, d≥1; n−1]], respectively.