Paper Info
Title
Quasi-Cyclic Codes Via Unfolded Cyclic Codes And Their Reversibility
Abstract
The finite field F-ql of V elements contains F-q as a subfield. If theta is an element of F-ql is of degree l over F-q, it can be used to unfold elements of F-ql to vectors in F-q(l). We apply the unfolding to the coordinates of all codewords of a cyclic code C over F(ql )of length n. This generates a quasi-cyclic code Q over F-q of length nl and index l. We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a generator polynomial g(x) of a cyclic code C, we present a formula for a generator polynomial matrix for the unfolded code Q. On the other hand, for any quasi-cyclic code Q with a reduced generator polynomial matrix G, we provide a necessary and sufficient condition on G that determines whether or not the code Q can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set T of the cyclic code C that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal generator polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.
Year
DOI
Venue
2019
10.1109/ACCESS.2019.2960569
IEEE ACCESS
Keywords
DocType
Volume
Cyclotomic cosets, finite fields, generator polynomial matrix, quasi-cyclic codes, reversible codes
Journal
7
ISSN
Citations 
PageRank 
2169-3536
0
0.34
References 
Authors
0
2
Name
Order
Citations
PageRank
Ramy Taki Eldin100.68
Hajime Matsui2188.14