Skew generalized quasi-cyclic codes

• Main Authors: Abualrub, Taher, Ezerman, Martianus Frederic, Seneviratne, Padmapani, Solé, Patrick
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2020
• Subjects: Science::Mathematics; Generalized Skew Quasi-cyclic Codes; Skew Polynomial Codes
• Online Access: http://www.twmsj.az/Archive.aspx?JournalName=Contents%20V.9,%20N.2,%202018; https://hdl.handle.net/10356/139559
• Institution: Nanyang Technological University

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.