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Journal of Algebra Combinatorics Discrete Structures and Applications

Self-orthogonal and quantum codes over chain rings

Maryam Bajalan, Mina Moeini, Bahattin Yıldız

Published May 6, 2024• 0 citations
Publisher

Abstract

In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field $\mathbb{F}_q$. We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes. Received: 19 September 2023 | Accepted: 18 January 2024

DOI
10.13069/jacodesmath.v11i2.304

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In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field Fq\mathbb{F}_qFq​. We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes. Received: 19 September 2023 | Accepted: 18 January 2024
Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The mmm-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the mmm-adic residue codes over the quotient ring Fq[v]⟨vs−v⟩\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}⟨vs−v⟩Fq​[v]​. We determine the idempotent generators of the mmm-adic residue codes over Fq[v]⟨vs−v⟩\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}⟨vs−v⟩Fq​[v]​. We obtain some parameters of optimal mmm-adic residue codes over Fq[v]⟨vs−v⟩\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}⟨vs−v⟩Fq​[v]​ with respect to Griesmer bound for rings. Furthermore, we derive a condition for mmm-adic residue codes over Fq[v]⟨vs−v⟩\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}⟨vs−v⟩Fq​[v]​ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing mmm-adic residue codes over Fq[v]⟨vs−v⟩\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}⟨vs−v⟩Fq​[v]​ and give some examples to illustrate our findings.
Recently, codes over some special finite rings especially chain rings have been studied. More recently, codes over finite non-chain rings have been also considered. Study on codes over such rings or rings in general is motivated by the existence of some special maps called Gray maps whose images give codes over fields. Quantum error-correcting (QEC) codes play a crucial role in protecting quantum information. The construction of quantum codes via classical codes over 2 F was first introduced by Calderbank and Shor [4] and Steane [13] in 1996. This method, known as CSS construction, has received a lot of attention and it has allowed to find many good quantum stabilizer codes. Later, construction of quantum codes over larger alphabets from classical linear codes over q F has shown by Ketkar et al. in [10]. One direction of the main research in quantum error correction codes is constructing quantum codes that have large minimum distances [9] for a given size and length. In [14], based on classical quaternary constacyclic linear codes, some parameters for quantum codes are obtained. In [8, 9], respectively based on classical negacyclic and constacyclic linear codes some parameters for quantum MDS codes are presented. In this work, we determine self-dual and self-orthogonal codes arising from constacyclic codes over the group ring(Zq[v]/G
In the paper, we consider a finite non-chain commutative ring [Formula: see text], where [Formula: see text]. We mainly study the construction of quantum codes from cyclic codes over [Formula: see text]. For this, we obtained self-orthogonal codes over [Formula: see text] as gray images of linear and cyclic codes over [Formula: see text], then these codes over [Formula: see text] correspond to a cyclic code over [Formula: see text] of odd length [Formula: see text] used to determine the parameters of the quantum codes.
(Θ,ΔΘ,a)(Θ, Δ_Θ, \mathbf{a})(Θ,ΔΘ​,a)-cyclic codes over Fql\mathbb{F}_q^lFql​ and their applications in the construction of quantum codes
In this article, for a finite field Fq\mathbb{F}_qFq​ and a natural number l,l,l, let R\mathcal{R}R denote the product ring Fql.\mathbb{F}_q^l.Fql​. Firstly, for an automorphism ΘΘΘ of R,\mathcal{R},R, a ΘΘΘ-derivation ΔΘΔ_ΘΔΘ​ of R\mathcal{R}R and for a unit a\mathbf{a}a in R,\mathcal{R},R, we study (Θ,ΔΘ,a)(Θ, Δ_Θ, \mathbf{a})(Θ,ΔΘ​,a)-cyclic codes over R.\mathcal{R}.R. In this direction, we give an algebraic characterization of a (Θ,ΔΘ,a)(Θ, Δ_Θ, \mathbf{a})(Θ,ΔΘ​,a)-cyclic code over R\mathcal{R}R, determine its generator polynomial, and find its decomposition over Fq.\mathbb{F}_q.Fq​. Secondly, we give a necessary and sufficient condition for a (Θ,0,a)(Θ, 0, \mathbf{a})(Θ,0,a)-cyclic code to be Euclidean dual-containing code over R.\mathcal{R}.R. Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over Fq\mathbb{F}_qFq​ as Gray images of (Θ,ΔΘ,a)(Θ, Δ_Θ, \mathbf{a})(Θ,ΔΘ​,a)-cyclic codes over R.\mathcal{R}.R. Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions.