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Mass Formula for Self-Orthogonal and Self-Dual Codes over Non-Unital Rings of Order Four

Author

Listed:
  • Adel Alahmadi

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

  • Altaf Alshuhail

    (Research Group of Algebraic Structures and Applications, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
    Department of Mathematics, Faculty of Science, University of Hail, Hail 55431, Saudi Arabia)

  • Rowena Alma Betty

    (Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines)

  • Lucky Galvez

    (Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines)

  • Patrick Solé

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

Abstract

We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring I = a , b | 2 a = 2 b = 0 , a 2 = b , ab = 0 and the noncommutative ring E = a , b | 2 a = 2 b = 0 , a 2 = a , b 2 = b , ab = a , ba = b . We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the formulas for the number of inequivalent self-orthogonal and self-dual codes, of a given type, over the said rings. Finally, using the mass formulas, we classify self-orthogonal and self-dual codes over each ring, for small lengths and types.

Suggested Citation

  • Adel Alahmadi & Altaf Alshuhail & Rowena Alma Betty & Lucky Galvez & Patrick Solé, 2023. "Mass Formula for Self-Orthogonal and Self-Dual Codes over Non-Unital Rings of Order Four," Mathematics, MDPI, vol. 11(23), pages 1-17, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:23:p:4736-:d:1285804
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    Cited by:

    1. Adel Alahmadi & Altaf Alshuhail & Rowena Alma Betty & Lucky Galvez & Patrick Solé, 2024. "The Mass Formula for Self-Orthogonal and Self-Dual Codes over a Non-Unitary Non-Commutative Ring," Mathematics, MDPI, vol. 12(6), pages 1-11, March.

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