Cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$
  Let $p$ be a prime number. In this paper, we study cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$. We find a unique set of generators for these codes. We also study the rank and the Hamming distance of these codes. We obtain all except one ternary optimal code of length 12 as the Gray image of the cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$. We also characterize the $p$-ary image of these cyclic codes under the Gray map. 
Pramod Kumar Kewat, Bappaditya Ghosh and Sukhamoy Pattanayak
cs.IT math.IT