Good coverings of Hamming spaces with spheres

The covering radius problem has been considered by many authors (e.g. [ 1, 5, 61). Finally, let t(n, k) be the minimum possible covering radius for an (n, k) code and k(n, p) the minimum possible dimension of a code with covering radius p. The study of t(n, k) was initiated by Karpovsky. For a survey of these questions, see ]41. The main goal of this paper is to find good linear coverings. The unrestricted (nonlinear) case is considered in Section 4, where existence theorems for coverings are given in a generalized setting, namely coverings of association schemes by tiles, using a result of Lovasz (based on the greedy algorithm [8]). Our first result is the following.