Bulletin of the
Korean Mathematical Society BKMS

Let $R_{u^2, v^2, w^2, p}$ be a finite non chain ring $ \mathbb{F}_p[u,v,w]\langle u^2,$ $v^2, w^2$, $uv-vu, vw-wv, uw-wu \rangle$, where $p$ is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u^2, v^2, w^2, p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length $8n$ over $\mathbb{F}_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.

Keywords: cyclic codes, Hamming distance, Gray map

MSC numbers: 94B15, 94B05