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

In this paper, we study cyclic codes over the ring R_p = F_p +uF_p + vF_p, where u² = v² = uv = vu = 0 (p: a prime number). We first derive generators of ideals of R_p[x]/〈xⁿ − 1〉, which are corresponding to cyclic codes over R_p 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_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 (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.