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IEEE Communications Letters

New Non-Binary Quantum Codes from Cyclic Codes Over Product Rings

Tushar Bag, Hai Quang Dinh, Ashish Kumar Upadhyay, Woraphon Yamaka

Published Dec 13, 2019• 25 citations
Publisher

Abstract

For any odd prime p, and a divisor ℓ of p, we consider Ie to be the set of all divisors of p - 1, which are less than or equal to ℓ. In this letter, we construct quantum codes from cyclic codes over Fpand FpSℓ, where Sℓ= Πi∈IℓRi, for Ri= (Fp[u])/(〈ui+1-u〉). For that, first we construct linear codes and a Gray map over Rℓ. Using this construction, we study cyclic codes over Rℓ, and then extend that over FpSℓ. We also give a Gray map over FpSℓ. Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature.

DOI
10.1109/lcomm.2019.2959529

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Asian-European Journal of Mathematics•Jul 27, 2018•19 citations
For any odd prime p, and a divisor ℓ of p, we consider Ie to be the set of all divisors of p - 1, which are less than or equal to ℓ. In this letter, we construct quantum codes from cyclic codes over Fpand FpSℓ, where Sℓ= Πi∈IℓRi, for Ri= (Fp[u])/(〈ui+1-u〉). For that, first we construct linear codes and a Gray map over Rℓ. Using this construction, we study cyclic codes over Rℓ, and then extend that over FpSℓ. We also give a Gray map over FpSℓ. Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature.
For an odd prime p and m ≥ 1, let R = Fpm[v]/{v3} = v}. Here our aim is to construct new non-binary quantum codes from skew constacyclic codes over this non-chain ring ℛ. In order to achieve our goal, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring ℛ. Then we determine a necessary and sufficient condition for these codes to contain their Euclidean duals. Finally, by applying CSS construction on these dual containing skew constacyclic codes and using Gray map, we obtain several new quantum codes.
Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.
In this paper, we extend the results given in [3] to a nonchain ring Rp=Fp+vFp+⋯+vp−1FpR_p={\mathbb{F}}_p+v{\mathbb{F}}_p+{\cdots}+v^{p-1}{\mathbb{F}}_pRp​=Fp​+vFp​+⋯+vp−1Fp​ , where vp=vv^p=vvp=v and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring RpR_pRp​ and study the structure of their duals. We classify cyclic codes containing their duals over RpR_pRp​ by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map π{\pi}π defined in [4], we give the parameters of the quantum codes of length pn over Fp{\mathbb{F}}_pFp​ which are obtained from cyclic codes over RpR_pRp​ . Finally, we illustrate the results by giving some examples.
(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.
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm\mathbb{F}_{p^m}Fpm​.