Lower bounds on covering codes via partition matrices

Let K-q(n, R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let sigma(q)(n, s: r) denote the minimal cardinality of a q-ary code of length n, which is S-SUr-jective with radius r. In order to lower-bound K-q (n, n - 2) and sigma(q) (n, s: s - 2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on K-q(n.n - 2) and to the new bound K-q(n.n - 2) >= 3q-2n + 2, improving the first iff 5 <= n < q <= 2n - 4. We determine K-q(q. q - 2) q - 2 + sigma(2)(q. 2: 0) if q <= 10. Moreover, we obtain the new powerful recursive bound Kq+i (n + 1, R + 1) >= min{2(q + 1), K-q (n, R) + 1}. (C) 2008 Elsevier Inc. All rights reserved.