IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i7p1099-d1370921.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/7/1099/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/7/1099/
    Download Restriction: no
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.

      Corrections

      All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1099-:d:1370921. See general information about how to correct material in RePEc.

      If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

      If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

      If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

      For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

      Please note that corrections may take a couple of weeks to filter through the various RePEc services.

      IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.