From skew-cyclic codes to asymmetric quantum codes

Abstract

We introduce an additive but not F4-linear map S from Fn4 To F24n and exhibit some of its interesting structural properties. If C is a linear [n, k, d]4-code, then S(C) is an additive (2n, 22k, 2d)4-code. If C is an additive cyclic code then S(C) is an additive quasi-cyclic code of index 2. Moreover, if C is a module θ-cyclic code, a recently introduced type of code which will be explained below, then S(C) is equivalent to an additive cyclic code if n is odd and to an additive quasi-cyclic code of index 2 if n is even. Given any (n, M, d)4-code C, the code S(C) is self-orthogonal under the trace Hermitian

inner product. Since the mapping S preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.