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.