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October 7, 2025Open Access

Cyclic, LCD, and Self-Dual Codes over the Non-Frobenius Ring GR(p2,m)u/⟨u2,pu⟩

Key Points

Cyclic codes over the non-Frobenius ring are systematically characterized through their generators and algebraic structure.
For gcd(n,p)=1, we provide an explicit description of the cyclic code generators, facilitating their enumeration.
Necessary and sufficient conditions for self-dual and LCD codes are established, highlighting critical aspects of cyclic code theory.
The paper includes numerous examples and tables that illustrate results, such as a mass formula for cyclic self-orthogonal codes.

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=GR(p2,m)u, where u2=0 and pu=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.

Cyclic, LCD, and Self-Dual Codes over the Non-Frobenius Ring GR(p2,m)u/⟨u2,pu⟩

Key Points

Abstract

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