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On Linear Codes over Finite Singleton Local Rings

Author

Listed:
  • Sami Alabiad

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • Alhanouf Ali Alhomaidhi

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • Nawal A. Alsarori

    (Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India)

Abstract

The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32 . To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z 4 codes, which play a significant role in coding theory.

Suggested Citation

  • Sami Alabiad & Alhanouf Ali Alhomaidhi & Nawal A. Alsarori, 2024. "On Linear Codes over Finite Singleton Local Rings," Mathematics, MDPI, vol. 12(7), pages 1-13, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1099-:d:1370921
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    References listed on IDEAS

    as
    1. Yousef Alkhamees & Sami Alabiad, 2022. "The Structure of Local Rings with Singleton Basis and Their Enumeration," Mathematics, MDPI, vol. 10(21), pages 1-10, October.
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