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Mathematical Sciences and Applications E-Notes

The Quantum Codes over F_q and Quantum Quasi-cyclic Codes over F_p

Yasemin Çengellenmiş, Abdullah Dertli

Published Apr 30, 2019• 7 citations
Publisher

Abstract

In this paper, the quantum codes over F q are constructed by using the cyclic codes over the finite ring R = F q + vF q + ... + v m-1 F q , where p is prime, q = p s , m -1|p -1 and v m = v.The parameters of quantum error correcting codes over F q are obtained.Some examples are given.Morever, the quantum quasi-cyclic codes over F p are obtained, by using the self dual basis for F p s over F p .

DOI
10.36753/mathenot.559260

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In this paper, the quantum codes over F q are constructed by using the cyclic codes over the finite ring R = F q + vF q + ... + v m-1 F q , where p is prime, q = p s , m -1|p -1 and v m = v.The parameters of quantum error correcting codes over F q are obtained.Some examples are given.Morever, the quantum quasi-cyclic codes over F p are obtained, by using the self dual basis for F p s over F p .
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 .