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

Sami Alabiad () and Alhanouf Ali Alhomaidhi
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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

Mathematics, 2025, vol. 13, issue 19, 1-17

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.

Keywords: cyclic code; self-dual code; LCD code; non-frobenius ring (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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