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arXiv:2412.13616

Quantum Codes from Group Codes

Kanat Abdukhalikov, Tushar Bag, Daniel Panario

Published Jan 1, 2024• 0 citations
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Abstract

This article examines group ring codes over finite fields and finite groups. We also present a section on two-dimensional cyclic codes in the quotient ring F q [x, y]/⟨x l -1, y m -1⟩. These two-dimensional cyclic codes can be analyzed using the group ring F q (C l × C m ), where C l and C m represent cyclic groups of orders l and m, respectively. The aim is to show that studying group ring codes provides a more compact approach compared to the quotient ring method. We further extend this group ring framework to study codes over other group structures, such as the dihedral group, direct products of cyclic and dihedral groups, direct products of two cyclic groups, and semidirect products of two groups. Additionally, we explore necessary and sufficient conditions for such group ring codes to be self-orthogonal under Euclidean, Hermitian, and symplectic inner products and propose a construction for quantum codes.

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This article examines group ring codes over finite fields and finite groups. We also present a section on two-dimensional cyclic codes in the quotient ring F q [x, y]/⟨x l -1, y m -1⟩. These two-dimensional cyclic codes can be analyzed using the group ring F q (C l × C m ), where C l and C m represent cyclic groups of orders l and m, respectively. The aim is to show that studying group ring codes provides a more compact approach compared to the quotient ring method. We further extend this group ring framework to study codes over other group structures, such as the dihedral group, direct products of cyclic and dihedral groups, direct products of two cyclic groups, and semidirect products of two groups. Additionally, we explore necessary and sufficient conditions for such group ring codes to be self-orthogonal under Euclidean, Hermitian, and symplectic inner products and propose a construction for 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.
Let Fq\mathbb{F}_qFq​ be a finite field of qqq elements, for some prime power qqq, and let GGG be a finite group. A (left) group code, or simply a GGG-code, is a (left) ideal of the group algebra Fq[G]\mathbb{F}_q[G]Fq​[G]. In this paper, we provide a complete group-algebraic description for the Hermitian dual code of any DnD_nDn​-code over Fq2\mathbb{F}_{q^2}Fq2​, where DnD_nDn​ is a dihedral group of order 2n2n2n with nnn not divisible by char⁡(Fq2)\operatorname{char}(\mathbb{F}_{q^2})char(Fq2​), through a suitable Wedderburn-Artin decomposition of the group algebra Fq2[Dn]\mathbb{F}_{q^2}[D_n]Fq2​[Dn​], and we determine all distinct Hermitian self-orthogonal DnD_nDn​-codes over Fq2\mathbb{F}_{q^2}Fq2​. We also present a thorough representation of the Euclidean dual code of any QnQ_nQn​-code over Fq\mathbb{F}_qFq​, where QnQ_nQn​ is a generalised quaternion group of order 4n4n4n not divisible by char⁡(Fq)\operatorname{char}(\mathbb{F}_q)char(Fq​), via the Wedderburn-Artin decomposition of the group algebra Fq[Qn]\mathbb{F}_q[Q_n]Fq​[Qn​]. In particular, since the semisimple group algebras Fq2[Qn]\mathbb{F}_{q^2}[Q_n]Fq2​[Qn​] and Fq2[D2n]\mathbb{F}_{q^2}[D_{2n}]Fq2​[D2n​] are isomorphic, then the Hermitian dual code of any QnQ_nQn​-code has also been fully described. As an application of the Hermitian dualities computed, we give a systematic construction, via the structure of the group algebra, to obtain quantum error-correcting codes, and in fact, with this methodical approach, we recover some already known quantum codes that achieve the best known minimum distance for their length and dimension.
In this article, for a finite field $ \mathbb{F}_q $ and a natural number $ l, $ let $ \mathcal{R} $ denote the product ring $ \mathbb{F}_q^l. $ Firstly, for an automorphism $ \Theta $ of $ \mathcal{R}, $ a $ \Theta $-derivation $ \Delta_\Theta $ of $ \mathcal{R} $ and for a unit $ \mathit{\boldsymbol{a}} $ in $ \mathcal{R}, $ we study $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic codes over $ \mathcal{R}. $ In this direction, we give an algebraic characterization of a $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic code over $ \mathcal{R} $, determine its generator polynomial, and find its decomposition over $ \mathbb{F}_q. $ Secondly, we give a necessary and sufficient condition for a $ (\Theta, 0, \mathit{\boldsymbol{a}}) $-cyclic code to be Euclidean dual-containing code over $ \mathcal{R}. $ Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over $ \mathbb{F}_q $ as Gray images of $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic codes over $ \mathcal{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.
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.