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Cyclic, LCD, and Self-Dual Codes over the Non-Frobenius Ring GR ( p 2 , m )[ u ]/〈 u 2 , pu 〉

Author

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  • 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)

Abstract

Let p be a prime number and m be a positive integer. In this paper, we investigate cyclic codes of length n over the local non-Frobenius ring R = G R ( p 2 , m ) [ u ] , where u 2 = 0 and p u = 0 . We first determine the algebraic structure of cyclic codes of arbitrary length n . For the case gcd ( n , p ) = 1 , we explicitly describe the generators of cyclic codes over R . Moreover, we establish necessary and sufficient conditions for the existence of self-dual and LCD codes, together with their enumeration. Several illustrative examples and tables are presented, highlighting the mass formula for cyclic self-orthogonal codes, cyclic LCD codes, and families of new cyclic codes that arise from our results.

Suggested Citation

  • Sami Alabiad & Alhanouf Ali Alhomaidhi, 2025. "Cyclic, LCD, and Self-Dual Codes over the Non-Frobenius Ring GR ( p 2 , m )[ u ]/〈 u 2 , pu 〉," Mathematics, MDPI, vol. 13(19), pages 1-17, October.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:19:p:3193-:d:1765342
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