Generator matrices of quasi-cyclic codes

BAYRAKTAR G. C., ÖZBUDAK F., ÖZKAYA B.

Discrete Mathematics, cilt.349, sa.11, 2026 (SCI-Expanded, Scopus) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 349 Sayı: 11
  • Basım Tarihi: 2026
  • Doi Numarası: 10.1016/j.disc.2026.115235
  • Dergi Adı: Discrete Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Applied Science & Technology Source, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO)
  • Anahtar Kelimeler: Generator matrix, Quasi-cyclic codes, Spectral analysis
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

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.