Maximal partial packings of Z2n with perfect codes

Abstract A maximal partial Hamming packing of Zn 2 is defined by us to be a family S of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in S. The number of translates of Hamming codes in S is the packing number, and a partial Hamming packing is strictly partial if the family S does not constitute a partition of Zn 2 . A simple and useful condition describing when two translates of Hamming codes are disjunct or not disjunct is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of Zn 2 , n = 2m − 1, satisfies m + 1 ≤ p ≤ n− 1. It is also proved that for any n equal to 2m−1, m ≥ 4, there exist maximal strictly partial Hamming packings of Zn 2 with packing numbers n−10, n−9, n−8, . . . , n−1. This implies that the upper bound is tight for any n = 2m − 1, m ≥ 4. All packing numbers for maximal strictly partial Hamming packings of Zn 2 , n = 7 and 15, are given by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5, 6, 7, . . . , 13 and 14.