Quasi-Cyclic Codes Via Unfolded Cyclic Codes and Their Reversibility

IEEE Access

Ramy Taki Eldin¹  Hajime Matsui² 

[1] Faculty of Engineering, Ain Shams University, Cairo, Egypt;Toyota Technological Institute, Nagoya, Japan;

关键词: Cyclotomic cosets;    finite fields;    generator polynomial matrix;    quasi-cyclic codes;    reversible codes;   

DOI  :  10.1109/ACCESS.2019.2960569

来源: DOAJ

【 摘 要 】

The finite field F_q(ℓ) of q^ℓ elements contains F_q as a subfield. If θ ∈ F_q(ℓ) is of degree ℓ over F_q, it can be used to unfold elements of F_q(ℓ) to vectors in F^ℓ_q. We apply the unfolding to the coordinates of all codewords of a cyclic code C over F_q(ℓ) of length n. This generates a quasi-cyclic code Q over Fq 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.

【 授权许可】

Unknown