Problems of Information Transmission, Vol. 41, No. 3, 2005, pp. 199–203. Translated from Problemy Peredachi Informatsii, No. 3, 2005, pp. 17–22.
Original Russian Text Copyright
2005 by Lebedev.
A Note on the Uniqueness of (w, r) Cover-Free Codes
V. S. Lebedev
Institute for Information Transmission Problems, Moscow
Received February 1, 2005; in ﬁnal form, May 5, 2005
Abstract—A binary code is called a (w, r) cover-free code if it is the incidence matrix of a
family of sets where the intersection of any w sets is not covered by the union of any other r sets.
For certain (w, r) cover-free codes with a simple structure, we obtain a new condition of opti-
mality and uniqueness up to row and/or column permutations.
Let us recall the deﬁnition of (w, r) cover-free codes. These codes have intensively been studied
since 1988 (see [1–4]).
Deﬁnition. An N×T binary matrix C is called a (w,r) cover-free code if for any pair of disjoint
⊂ [T ] of cardinalities |J
| = w and |J
| = r there exists i ∈ [N] such that c
all j ∈ J
=0forallj ∈ J
Note that (w, r) cover-free codes exist only if T ≥ w + r. Moreover, inversion of all entries of
acodematrixC transposes the parameters w and r. Therefore, we assume throughout the paper
that w ≤ r.
We will often refer to columns of C as codewords and use the term code of size N × T instead of
themorecommonlyusedcode of length N and cardinality T .
The main problem in the study of (w, r) cover-free codes is to ﬁnd the maximum number of
codewords T (N,w,r) for a given length N, or to ﬁnd the minimal length N(T,w,r)ofacodefor
a given cardinality T .
We say that a (w, r)cover-freecodeofsizeN × T is optimal if N = N(T,w,r). In the case
r ≥ w ≥ 2, only a few examples of optimal (w, r) cover-free codes are known.
Thus, it was proved in [3, 4] that N(8, 2, 2) = 14; some other optimal cover-free codes were
obtained in : N (9, 2, 2) = 18, N(10, 2, 3) = 30, and N(11, 3, 3) = 66.
We say that two (w, r) cover-free codes are equivalent if the incidence matrix of one code can be
obtained from the incidence matrix of the other by a sequence of row and/or column permutations
(in the case w = r, we are also allowed to take inversion of all entries of the code matrix).
We say that a (w, r)cover-freecodeofsizeN × T is unique if it is equivalent to any other (w, r)
cover-free code of the same size N × T .
The optimality of the (2, 2) cover-free code of size 14 × 8 and the optimality of the (2, 3) cover-
free code of size 30 × 10 were proved in . In [7, 8] it was proved that the (2, 2) cover-free code
of size 18 × 9 is unique, and in  it was proved that the (3, 3) cover-free code of size 66 × 11 is
Thus, all known optimal and nontrivial (w, r) cover-free codes were proved to be unique. In the
subsequent sections we discuss what trivial codes are and consider their optimality and uniqueness.
Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00098.
2005 Pleiades Publishing, Inc.