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Cyclic Codes over a Non-Local Non-Unital Ring

Author

Listed:
  • Adel Alahmadi

    (Research Group of Algebraic Structures and Applications, Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Malak Altaiary

    (Research Group of Algebraic Structures and Applications, Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
    Math Department, Shaqra University, Shaqra 11961, Saudi Arabia)

  • Patrick Solé

    (I2M Lab (CNRS, Aix Marseille University, Centrale Marseille), 13009 Marseilles, France)

Abstract

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H = ⟨ a , b ∣ 2 a = 2 b = 0 , a 2 = 0 , b 2 = b , a b = b a = 0 ⟩ . This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n . We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H . We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2 n over F 2 is studied.

Suggested Citation

  • Adel Alahmadi & Malak Altaiary & Patrick Solé, 2024. "Cyclic Codes over a Non-Local Non-Unital Ring," Mathematics, MDPI, vol. 12(6), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:6:p:866-:d:1357771
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