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QUANTUM CODES FROM CYCLIC CODES OVER $\mathbb{F}_{q}+u \mathbb{F}_{q}+v \mathbb{F}_{q}$

Peng Hu, Hui Li, Xiusheng Liu

Published Mar 15, 2021• 0 citations
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Abstract

In this paper, we investigate the method of constructing quantum codes from cyclic codes over the ring $R= \mathbb{F}_{q}+u \mathbb{F}_{q}+v \mathbb{F}_{q}$ (u2=u, v2=v, uv=vu=0). By means of the generator polynomials and some decomposition of cyclic codes and their dual over R, we give a necessary and sufficient condition of the cyclic codes over R to be construct the quantum codes. Also as an application, we obtain new non binary quantum codes from those classes of cyclic codes.

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QUANTUM CODES FROM CYCLIC CODES OVER Fq+uFq+vFq\mathbb{F}_{q}+u \mathbb{F}_{q}+v \mathbb{F}_{q}Fq​+uFq​+vFq​
In this paper, we investigate the method of constructing quantum codes from cyclic codes over the ring R=Fq+uFq+vFqR= \mathbb{F}_{q}+u \mathbb{F}_{q}+v \mathbb{F}_{q}R=Fq​+uFq​+vFq​ (u2=u, v2=v, uv=vu=0). By means of the generator polynomials and some decomposition of cyclic codes and their dual over R, we give a necessary and sufficient condition of the cyclic codes over R to be construct the quantum codes. Also as an application, we obtain new non binary quantum codes from those classes of cyclic codes.
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.
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 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.
On the Structure of Cyclic Codes Over FqRS\mathbb{F}_{q}{RS}Fq​RS and Applications in Quantum and LCD Codes Constructions
Let p be an odd prime, q = pm, R = Fq+ uFqwith u2= 1, and S = Fq+ uFq+ vFq+ uvFqwith u2= 1, v2= 1, uv = vu. In this paper, FqRS-cyclic codes over FqRS are studied. As an application, we present a construction of quantum error-correcting codes (QECCs) from the FqRS-cyclic codes over FqRS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from the FqRS-cyclic codes over FqRS. Among others, we construct a Gray map over FqRS and discuss the Gray images of FqRS-cyclic codes over Fq.