On Quantum Codes From Cyclic Codes Over a Class of Nonchain Rings

Abstract

Abstract. In this paper, we extend the results given in [3] to a nonchain ring Rp = Fp + vFp + · · · + v p−1 Fp, where v p = v and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring Rp and study the structure of their duals. We classify cyclic codes containing their duals over Rp by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map π defined in [4], we give the parameters of the quantum codes of length pn over Fp which are obtained from cyclic codes over Rp. Finally, we illustrate the results by giving some examples.

A Study of FqR-Cyclic Codes and Their Applications in Constructing Quantum Codes

“…Calderbank et al [17] subsequently studied the construction of this family of codes from classical codes, demonstrated their existence and methods of correction. In the last few years, a number of QECCs were discussed over finite fields using the theory of Calderbank et al [17] (See [2], [3], [25], [27], [28], [32]- [37], [40], [41]).…”

New quantum codes from constacyclic codes over the ring $ R_{k,m} $

“…Meantime, Dertli et al [8] presented some new binary quantum codes obtained from the cyclic codes over F 2 + uF 2 + vF 2 + uvF 2 , and then Ashraf and Mohammad [2] generalized their work over the ring F q + uF q + vF q + uvF q to derive new non-binary quantum codes. There are a lot of articles in which good quantum codes are obtained from the cyclic codes on different finite rings, see [11,13,19,23,31,33,32,34]. On the other side, recently, Gao and Wang [12], Li et al [27], Ma et al [29,30] considered the constacyclic codes over finite non-chain rings and obtained many new and better codes compare to the known codes.…”

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

“…Many good quantum codes has been constructed by using the classical cyclic codes over F q with self orthogonal (or dual containing) properties. Recently, some authors have constructed quantum the codes by using the linear codes over some finite ring in [1][2][3][4][8][9][10][12][13][14][15][16][17][18].…”

“…In 2015, Gao constructed the quantum codes over F q from the cyclic codes over a finite non chain ring F q + vF q + v 2 F q + v 3 F q , where q = p r , p is an odd prime, 3|p − 1 and v 4 = v in [8]. In 2016, Sari and Siap constructed the quantum codes over F p from the cyclic codes of arbitrary length over F p + vF p + ... + v p−1 F p , where v p = v and p is a prime in [13].…”