Further results on the covering radius of small codes

The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K (t, b, R). In the paper, necessary and sufficient conditions for K (t, b, R) = M are given for M = 6 and 7 by proving that there exist exactly three families of optimal codes with six codewords and two families of optimal codes with seven codewords. The cases M <= 5 were settled in an earlier study by the same authors. For binary codes, it is proved that K (0, 2b + 4, b) >= 9 for b >= 1. For ternary codes, it is shown that K (3t + 2, 0, 2t) = 9 for 1 >= 2. New upper bounds obtained include K (3t + 4, 0, 2t) <= 36 for t <= 2. Thus, we have K (13, 0, 6) <= 36 (instead of 45, the previous best known upper bound). (c) 2006 Elsevier B.V. All rights reserved.