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Peer-reviewedCC BY 4.0Entropy

Cyclic Codes over a Commutative Non-Unital Ring

  • Adel Alahmadi
  • Tamador Alihia
  • Malak Altaiary
  • Abdul Nadim Khan
  • Patrick SolΓ©
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Abstract

This paper investigates cyclic codes over the commutative non-unital ring 𝕀 = ⟨ a, B | 2a = 2B = 0, a2 = B, ab = 0 ⟩, a ring of order four and characteristic two. We develop a structural description of linear and cyclic codes over 𝕀, analyze their residue and torsion codes, and characterize self-dual codes. A Gray map from 𝕀n to 𝔽2n is introduced and shown to transform cyclic codes into binary quasi-cyclic codes of index two. Numerical examples for lengths 2 ≀ n ≀ 7 illustrate the diversity of cyclic codes over 𝕀 and highlight their potential applications.

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Cite

BibTeXRIS
BibTeX
@article{alahmadi2026cyclic,
  title = {{Cyclic Codes over a Commutative Non-Unital Ring}},
  author = {Adel Alahmadi and Tamador Alihia and Malak Altaiary and Abdul Nadim Khan and Patrick SolΓ©},
  journal = {Entropy},
  year = {2026},
  month = oct,
  doi = {10.3390/e28111185},
  url = {https://hal.science/hal-05861490v1},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Cyclic Codes over a Commutative Non-Unital Ring
AU  - Adel Alahmadi
AU  - Tamador Alihia
AU  - Malak Altaiary
AU  - Abdul Nadim Khan
AU  - Patrick SolΓ©
PY  - 2026
DA  - 2026/10/27
JO  - Entropy
DO  - 10.3390/e28111185
UR  - https://hal.science/hal-05861490v1
AB  - This paper investigates cyclic codes over the commutative non-unital ring 𝕀 = ⟨ a, B | 2a = 2B = 0, a^2 = B, ab = 0 ⟩, a ring of order four and characteristic two. We develop a structural description of linear and cyclic codes over 𝕀, analyze their residue and torsion codes, and characterize self-dual codes. A Gray map from 𝕀^n to 𝔽_2^n is introduced and shown to transform cyclic codes into binary quasi-cyclic codes of index two. Numerical examples for lengths 2 ≀ n ≀ 7 illustrate the diversity of cyclic codes over 𝕀 and highlight their potential applications.
SN  - 1099-4300
ER  -