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- Title
Various structures of cyclic codes over the non-Frobenius ring $\mathbb{F}_p[u, v] /\left\langle u^2, v^2, u v, v u\right\rangle$.
- Authors
Choi, Hyun Seung; Kim, Boran
- Abstract
In this paper, we study cyclic codes over the ring $ R_{p} = \mathbb{F}_{p}+u\mathbb{F}_{p}+v\mathbb{F}_{p} $, where $ u^2 = v^2 = uv = vu = 0 $ ($ p $: a prime number). We first derive generators of ideals of $ R_{p}[x]/\langle x^n-1\rangle $, which are corresponding to cyclic codes over $ R_{p} $ of arbitrary length $ n $. Especially, for $ \text{gcd}(n,p) = 1 $, we have the explicit generators of ideals corresponding to cyclic codes, their duals, self-orthogonal codes and self-dual codes over $ R_{p} $ of length $ n $. Furthermore, mass formulae of cyclic self-orthogonal and LCD codes over $ R_{p} $ of length $ n $ is obtained. Finally, we present a Gray map from $ (R_{p})^{n} $ to $ (\mathbb{F}_{p})^{6n} $, showing that the image of a cyclic code over this Gray map is quasi-cyclic, and determine the index of this image. A series of examples and tables concerning the mass formula of cyclic self-orthogonal codes, cyclic LCD codes and several new quasi-cyclic codes are presented as an application of theorems.
- Subjects
CYCLIC codes; PRIME numbers; GRAY codes; HYPERGEOMETRIC series
- Publication
Advances in Mathematics of Communications, 2024, Vol 18, Issue 2, pN.PAG
- ISSN
1930-5346
- Publication type
Article
- DOI
10.3934/amc.2023030