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Journal of Discrete Mathematical Sciences and Cryptography

Improved and Novel Quantum Codes through CSS Construction from Cyclic Codes over Commutative Non-Local Ring Fp×(Fp + αFp + α2 Fp)

Arun Kumar Yadav, Manju Pruthi

Published Jan 1, 2024• 1 citation
Publisher

Abstract

In this study, by using CSS construction, we present the new quantum codes derived from cyclic codes over the commutative non - local ring R = Fp × (Fp + αFp + α2 Fp), where p is an odd prime, α3 = α. Firstly, we investigate the structure of the ring R and study linear codes by defining Lee weight over R. We define a distance – preserving Gray map from R→Fp 4 and discuss the cyclic codes with their dual code by Euclidean inner products on R. Additionally, we find the quantum Singleton defect(QSD) of the quantum code, which is denoted as n + 2 – k – 2d. Magma Software will be employed to provide relevant illustrations for enhanced comprehension.

DOI
10.47974/jdmsc-2001

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In this study, by using CSS construction, we present the new quantum codes derived from cyclic codes over the commutative non - local ring R = Fp × (Fp + αFp + α2 Fp), where p is an odd prime, α3 = α. Firstly, we investigate the structure of the ring R and study linear codes by defining Lee weight over R. We define a distance – preserving Gray map from R→Fp 4 and discuss the cyclic codes with their dual code by Euclidean inner products on R. Additionally, we find the quantum Singleton defect(QSD) of the quantum code, which is denoted as n + 2 – k – 2d. Magma Software will be employed to provide relevant illustrations for enhanced comprehension.
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.
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.
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.
The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R=Fp+w1Fp+w2Fp+w22Fp+w1w2Fp+w1w22Fp, where w12−α2=0, w1w2=w2w1, w23−β2w2=0, and α,β∈Fp∖{0}, for a prime p. We begin by introducing a Gray map defined over R, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction.
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​.