Cyclic codes over the ring ${\mathbb F}_p[u,v,w]/\langle u^2, v^2, w^2, uv-vu, vw-wv, uw-wu \rangle$
Bull. Korean Math. Soc. 2018 Vol. 55, No. 1, 115-137
Published online January 1, 2018
Pramod Kumar Kewat, Sarika Kushwaha
Indian Institute of Technology (ISM), Indian Institute of Technology (ISM)
Abstract : 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
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