This paper gives lower and upper bounds on the covering radius of codes over Z 2 s with respect to homogenous distance. We also determine the covering radius of various Repetition codes, Simplex codes (Type α and Type β) and their dual and give bounds on the covering radii for MacDonald codes of both types over Z 4 .

Let p be an odd prime and q = p m , where m is a positive integer. We study the Θ t -cyclic and (Θ t , λ )-cyclic code over a finite commutative non-chain ring R = F q [u, v, w]/ u 2 = u, v 2 = v, w 2 = 1, uv = vu = 0, uw = wu, wv = vw , where λ is a unit in R.

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Lower and upper bounds for R(S k (q)) , the covering radius of a k -dimensional q -ary Simplex codes are determined. These help in getting bounds with a gap of one for S 3 (q) . Exact covering radius of S 2 (q) , S 3 (3) , S 4 (3) , S 4 (4) and S 3 (q) for q even are obtained.