Abstract.

This article discusses skew generalized quasi-cyclic codes over any finite field F with Galois automorphism θ. This is a generalization of both quasi-cyclic codes and skew polynomial codes. These codes have an added advantage over quasi-cyclic codes since their lengths do not have to be multiples of the index. After a brief description of the skew polynomial ring F[x; θ], we show that a skew generalized quasi-cyclic code C is a left submodule of R1×R2×. . .×Rℓ, where Ri , F[x; θ]/(xmi −1), with |⟨θ⟩| = m and m divides mi for all i ∈ {1, . . . , ℓ}. This description provides a direct construction of many codes with best-known parameters over GF(4). As a byproduct, some good asymmetric quantum codes detecting single bit-flip error can be derived from the constructed codes.

Keywords:

generalized skew quasi-cyclic codes, skew polynomial codes, quasi-cyclic codes, quantum CSS codes.