Skew Generalized Quasi-cyclic Codes

This article discusses skew generalized quasi-cyclic codes over any finite field F with Galois automorphism θ. This is a generalization of 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× . . .×R`, where Ri , F[x; θ]/(xi −1), with |〈θ〉| = m and m divides mi for all i ∈ {1, . . . , `}. 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.