VARIOUS STRUCTURES OF CYCLIC CODES OVER THE NON-FROBENIUS RING Fp [u, v]/〈u2, v2, uv, vu〉

초록

In this paper, we study cyclic codes over the ring R<inf>p</inf> = F<inf>p</inf> +uF<inf>p</inf> + vF<inf>p</inf>, where u2 = v2 = uv = vu = 0 (p: a prime number). We first derive generators of ideals of R<inf>p</inf>[x]/〈xn − 1〉, which are corresponding to cyclic codes over R<inf>p</inf> of arbitrary length n. Especially, for 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<inf>p</inf> of length n. Furthermore, mass formulae of cyclic self-orthogonal and LCD codes over R<inf>p</inf> of length n is obtained. Finally, we present a Gray map from (R<inf>p</inf>)n to (F<inf>p</inf>)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. © 2024, American Institute of Mathematical Sciences. All rights reserved.

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