# From Skew-Cyclic Codes to Asymmetric Quantum Codes

Martianus Frederic Ezerman, San Ling, Patrick Sole, Olfa Yemen

Arxiv ID: 1005.0879•Last updated: 4/28/2020

We introduce an additive but not 𝔽_4-linear map S from 𝔽_4^n to 𝔽_4^2n 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,2^2k,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.

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