On the Construction of Skew Quasi-Cyclic Codes
Taher Abualrub, Ali Ghrayeb, Nuh Aydın, İrfan Şiap · IEEE Transactions on Information Theory · 2010
In this paper, we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a noncommutative ring called the skew polynomial ringF[x;¿]. After a brief description of the skew polynomial ringF[x;¿], it is shown that skew QC codes are left submodules of the ringRsl=(F[x;¿]/(xs-1) )l. The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ringF[x;¿] and show that parity-check polynomials for skew QC codes are unique up to similarity. Our search results lead to the construction of several new codes with Hamming distances exceeding the Hamming distances of the previously best known linear codes with comparable parameters.