$${\mathbb {F}}_qR$$-linear skew constacyclic codes and their application of constructing quantum codes

Abstract

Let q be a prime power with $$\mathrm{gcd}(q,6)=1$$ . Let $$R={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+v{\mathbb {F}}_{q^2}+uv{\mathbb {F}}_{q^2}$$ , where $$u^2=u$$ , $$v^2=v$$ and $$uv=vu$$ . In this paper, we give the definition of linear skew constacyclic codes over $${\mathbb {F}}_{q^2}R$$ . By the decomposition method, we study the structural properties and determine the generator polynomials and the minimal generating sets of linear skew constacyclic codes. We define a Gray map from $${\mathbb {F}}_{q^2}^{\alpha }\times R^{\beta }$$ to $${\mathbb {F}}_{q^2}^{\alpha +4\beta }$$ preserving the Hermitian orthogonality, where $$\alpha $$ and $$\beta $$ are positive integers. As an application, by Hermitian construction, we obtain some good quantum error-correcting codes.