Problems of Information Transmission, Vol. 40, No. 3, 2004, pp. 195–201. Translated from Problemy Peredachi Informatsii, No. 3, 2004, pp. 13–20.
Original Russian Text Copyright
2004 by Kim, Lebedev.
On the Optimality of Trivial (w, r)-Cover-Free Codes
H. K. Kim
and V. S. Lebedev
Pohang University of Science and Technology, Korea
Institute for Information Transmission Problems, RAS, Moscow
Received December 23, 2003; in ﬁnal form, May 24, 2004
Abstract—A(w, r)-cover-free code is the incidence matrix of a family of sets where no in-
tersection of w members of the family is covered by the union of r others. We obtain a new
condition in view of which (w, r)-cover-free codes with a simple structure are optimal. We also
introduce (w, r)-cover-free codes with a constraint set.
Assume that we want to ﬁnd a minimal set of binary vectors of length 8 with the property that
the projection of this set onto any four coordinates i
contains a subvector with 1 in i
and 0 in i
. How many vectors do we need? The answer is 14. We can, for example, use the
vectors of weight 4 from the extended Hamming code:
It is known that no set of 13 binary vectors of length 8 possesses this property [1, 2]. Thus,
binary matrix (1) gives a minimal set. Moreover, one can prove that this set is unique up to row
or column permutations (see ).
Consider a more general situation.
Deﬁnition 1. AbinaryN × T matrix C = c
is called a (w, r)-cover-free code if, for any
pair of disjoint subsets J
⊂ [T ] of cardinalities |J
| = w and |J
| = r, there exists a coordinate
i ∈ [N] such that c
=1forallj ∈ J
=0forallj ∈ J
Supported in part by Com
Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00098, and NSF, Grant
2004 MAIK “Nauka/Interperiodica”