Generator matrices of quasi-cyclic codes
Discrete Mathematics, vol.349, no.11, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 349 Issue: 11
- Publication Date: 2026
- Doi Number: 10.1016/j.disc.2026.115235
- Journal Name: Discrete Mathematics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Applied Science & Technology Source, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO)
- Keywords: Generator matrix, Quasi-cyclic codes, Spectral analysis
- Middle East Technical University Affiliated: Yes
Abstract
In this paper, we provide a complete solution to an open problem by applying the spectral method to the polynomial generator matrices of quasi-cyclic codes of arbitrary index, using the corresponding reduced Gröbner basis of the given quasi-cyclic code. More precisely, with this method, we construct a full generator matrix over the suitable extension field. We provide two explicit examples to illustrate the construction of a full generator matrix from the given polynomial generator matrix of the code in the cases of index 2 and index 3, respectively. Furthermore, we investigate the reduced Gröbner basis of the subclass of 2D-cyclic codes of index 2 and index 3, proving that they have polynomial generator matrices in diagonal form.