Abstract
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.
Highlights
Quantum error-correcting codes have experienced tremendous growth since the discovery that there exists quantum error-correcting codes which protect quantum information as classical error-correcting codes protect classical information
Ashraf and Mohammad made many outstanding contributions in quantum code [5,6]. One of their contributions is that they constructed some new non-binary quantum codes over Fq + uFq + vFq + uvFq with u2 = v2 = 0
On the basis of previous studies, many other scholars have made great contributions of quantum codes from cyclic codes over different rings [3, 8,9,10]. These results show that some good quantum codes can be obtained by classical codes over finite rings
Summary
Quantum error-correcting codes have experienced tremendous growth since the discovery that there exists quantum error-correcting codes which protect quantum information as classical error-correcting codes protect classical information. On the basis of previous studies, many other scholars have made great contributions of quantum codes from cyclic codes over different rings [3, 8,9,10] These results show that some good quantum codes can be obtained by classical codes over finite rings. Designing the effective Gray map preserving self-orthogonal properties from rings to finite fields is crucial for constructing quantum codes by codes over rings. Some examples under this Gray map are recomputed and get some better quantum codes In this sense, this Gray map is more effective to produce new or good non-binary quantum codes.
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