# CYCLIC CODES OVER THE RING 𝔽p[u, v, w]/〈u2, v2, w2, uv - vu, vw - wv, uw - wu〉

• Kewat, Pramod Kumar (Department of Applied Mathematics Indian Institute of Technology (ISM)) ;
• Kushwaha, Sarika (Department of Applied Mathematics Indian Institute of Technology (ISM))
• Accepted : 2017.03.07
• Published : 2018.01.31

#### 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.

#### File

Table 1. Non zero cyclic codes of length 4 over Ru2,v2,w2,2.

Table 2. Non zero cyclic codes of length 3 over Ru2,v2,w2,3.

Table 3. Non zero cyclic codes of length 5 over Ru2,v2,w2,5.

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