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Erschienen in:
21.07.2018 | Original Paper
Quantum codes from cyclic codes over the ring \({\mathbb {F}}_q+v_1{\mathbb {F}}_q+\cdots +v_r{\mathbb {F}}_q\)
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 2/2019
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Abstract
Let \(R = {{\mathbb {F}}_q} + {v_1}{{\mathbb {F}}_q} + \cdots + {v_r}{{\mathbb {F}}_q},\) where q is a power of a prime, \(v_i^2=v_i,\; v_iv_j=v_jv_i=0\) for \(1\le i,j \le r\) and \(r\ge 1\). In this paper, the structure of cyclic codes over the ring R is studied and a Gray map \(\phi \) from \({R^n}\) to \({\mathbb {F}}_q^{(r + 1)n}\) is given. We give a construction of quantum codes from cyclic codes over the ring R. We derive Euclidean dual containing codes over \({\mathbb {F}}_q\) and Hermitian dual containing codes over \({\mathbb {F}}_{p^{2m}}\) as Gray images of cyclic codes over R. In particular, we use \(r+1\) codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code. Furthermore, some new non-binary quantum codes are obtained.