Skew generalized quasi-cyclic codes - Abualrub, Taher (2020)
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.