SACS | Classification of Certain Cyclic LCD Codes

Published in Volume XXXIV, Issue 1, 2024, pages 23-38, doi: 10.47743/SACS.2024.1.23

Authors: S. Gannon, H. Kulosman

Abstract

We show that a necessary and sufficient condition for a cyclic code C of length N over a finite chain ring R (whose maximal ideal has nilpotence 2) to be an LCD code is that C = (f(x)), where f(X) is a self-reciprocal monic divisor of X^N − 1 in R[X] and x = X + (X^N − 1) in R[X]/(X^N − 1). A similar, but slightly different, theorem was proved in 2019 by Z. Liu and J. Wang for general finite chain rings (Theorem 25 in [5]). We provide two proofs, both completely different than the proof of Liu and Wang.

Full Text (PDF)

References

[1] Yilmaz Durgun. On LCD codes over finite chain rings. Bulletin of the Korean Mathematical Society, 57(1):37–50, 2020. doi:10.4134/BKMS.b181173.

[2] Seth Gannon and Hamid Kulosman. The condition for a cyclic code over Z4 of odd length to have a complementary dual. CoRR, abs/1905.12309, 2019. arXiv:1905.12309.

[3] Yan Jia, San Ling, and Chaoping Xing. On self-dual cyclic codes over finite fields. IEEE Transactions on Information Theory, 57(4):2243–2251, 2011. doi:10.1109/TIT.2010.2092415.

[4] Somphong Jitman, Ekkasit Sangwisut, and Patanee Udomkavanich. Hulls of cyclic codes over Z4 . Discrete Mathematics, 343(1):111621, 2020. doi:10.1016/j.disc.2019.111621.

[5] Zihui Liu and Jinliang Wang. Linear complementary dual codes over rings. Designs, Codes and Cryptography, 87(12):3077–3086, 2019. doi:10.1007/S10623-019-00664-3.

[6] James L Massey. Linear codes with complementary duals. Discrete Mathematics, 106-107:337–342, 1992. doi:10.1016/0012-365X(92)90563-U.

[7] Graham H Norton and Ana Sălăgean. On the structure of linear and cyclic codes over a finite chain ring. Applicable algebra in engineering, communication and computing, 10(6):489–506, 2000. doi:10.1007/PL00012382.

[8] Xiang Yang and James L. Massey. The condition for a cyclic codes to have a complementary dual. Discrete Mathematics, 126(1-3):391–393, 1994. doi:10.1016/0012-365X(94)90283-6.

Bibtex

@article{sacscuza:gannon2024ccclc,
  title={Classification of Certain Cyclic LCD Codes},
  author={S. Gannon, H. Kulosman},
  journal={Scientific Annals of Computer Science},
  volume={34},
  number={1},
  organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania},
  year={2024},
  pages={23-38},
  publisher={Alexandru Ioan Cuza University Press, Ia\c si},
  doi={10.47743/SACS.2024.1.23}
}