On the construction of skew quasi-cyclic codes

Authors:
Taher Abualrub;Ali Ghrayeb;Nuh Aydin;Irfan Siap
Affiliations:
Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE;Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada;Department of Mathematics, Kenyon College, Gambier, OH;Department of Mathematics, Yildiz Technical University, Istanbul, Turkey
Venue:
IEEE Transactions on Information Theory
Year:
2010

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Abstract

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 ring F[x;θ]. After a brief description of the skew polynomial ring F[x;θ], it is shown that skew QC codes are left submodules of the ring Rs1 = (F[x;θ]/(xs - 1))1. The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ring F[x;θ] 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.