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Journal of Physics Conference Series

Quantum codes from cyclic codes over F3+ μF3+ υF3+ μυ F3

Mehmet Özen, N. Tuğba Özzaim, Halit İnce

Published Oct 1, 2016• 9 citations
Publisher

Abstract

In this paper, it will be argued that structure of cyclic codes over F 3 + μF 3 + υF 3 + μυ F 3 where u 2 = 1, v 2 = 1 and uv = vu for arbitrary length n. We define a new Gray map which is a distance preserving map. By using decomposition theory, we find generator polynomials of cyclic codes over F 3 + μF 3 + υF 3 + μυ F 3 and obtain the result that cyclic codes over F 3 + μF 3 + υF 3 + μυ F 3 are principally generated. Further, using these results we determine the parameters of quantum codes which constructed from cyclic codes over F 3 + μF 3 + υF 3 + μυ F 3 . We present some of the results of computer search.

DOI
10.1088/1742-6596/766/1/012020

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In this paper, it will be argued that structure of cyclic codes over F 3 + μF 3 + υF 3 + μυ F 3 where u 2 = 1, v 2 = 1 and uv = vu for arbitrary length n. We define a new Gray map which is a distance preserving map. By using decomposition theory, we find generator polynomials of cyclic codes over F 3 + μF 3 + υF 3 + μυ F 3 and obtain the result that cyclic codes over F 3 + μF 3 + υF 3 + μυ F 3 are principally generated. Further, using these results we determine the parameters of quantum codes which constructed from cyclic codes over F 3 + μF 3 + υF 3 + μυ F 3 . We present some of the results of computer search.
Let R 2 denotes the ring F 2 + μF 2 + υ 2 + μυ F 2 + wF 2 + μwF 2 + υwF 2 + μυwF 2 . In this study, we construct quantum codes from cyclic codes over the ring R 2 , for arbitrary length n, with the restrictions μ 2 = 0, υ 2 = 0, w 2 = 0, μυ = υμ, μw = wμ, υw = wυ and μ (υw) = (μυ) w. Also, we give a necessary and sufficient condition for cyclic codes over R 2 that contains its dual. As a final point, we obtain the parameters of quantum error-correcting codes from cyclic codes over R 2 and we give an example of quantum error-correcting codes form cyclic codes over R 2 .
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.
The structural properties and construction methodologies of quantum codes (QCs) defined over the finite field Z_3, the finite field with three elements, are examined in this work. The primary focus is on utilizing constacyclic codes within a specific algebraic structure the finite commutative non-chain ring R=Z3+uZ3+vZ3+uvZ3, where the indeterminates u and v satisfy the relations u2=1, v2=1, and uv=vu. We provide a set of idempotent generators of the ring and define linear codes and calculated some self-inverse units. The relationship between R and Z34 is established using a gray map. Constacyclic (CC) codes are broken down into cyclic (C) codes and negacyclic (N) codes in order to determine the parameters of QCs over Z_3. Several QCs of arbitrary lengths are developed as an application. In this paper, we calculated those various parameters of QCs which are better than existing parameters of QCs. This paper contains only those units which have self-inverse.
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.