Problems of Information Transmission, Vol. 39, No. 4, 2003, pp. 317–323. Translated from Problemy Peredachi Informatsii, No. 4, 2003, pp. 3–9.
Original Russian Text Copyright
2003 by Lebedev.
INFORMATION THEORY AND CODING THEORY
Asymptotic Upper Bound for the Rate
of (w, r) Cover-Free Codes
V. S. Lebedev
Institute for Information Transmission Problems, RAS, Moscow
Received October 1, 2002; in ﬁnal form, February 7, 2003
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 of the sets is not covered by the union of
any other r sets. Such a family is called a (w, r) cover-free family. We obtain a new recurrent
inequality for the rate of (w, r) cover-free codes, which improves previously known upper bounds
on the rate.
One of the most important applications of (w, r) cover-free codes is the following cryptography
problem. There are T users and N secret keys. Each user has his own set of keys, and a group of
users can communicate if there exists a common secret key for the whole group. It is required that,
for any group of w users and any group of r other users, there should exist a key such that all users
of the ﬁrst group have this key and thus can communicate, while neither of the r users of the second
group possess this key. Thus, users of the ﬁrst group can exchange information “secretly” from
users of the second group. This situation can naturally be thought of as a binary N × T matrix
C = c
=1ifthejth user possesses the ith key, and c
=0ifthejth user does not
have the ith key. Then the property described above means that, for any subsets J
⊂ [T ]of
| = w and |J
| = r,thereexistsarowi in C such that c
=1foranyj from J
=0foranyj from J
Of course, we would like to minimize the number of secret keys with a ﬁxed number of users,
or, equivalently, maximize the number of users with a ﬁxed number of keys. Thus, the problem
consists in ﬁnding a matrix C that obeys this property, with the number of columns as large as
possible(rowsareoflengthN ). We will often refer to columns of C as codewords and refer to
the matrix C itself as a binary code. Furthermore, in what follows, we use the term “code of size
N × T ” rather than the more commonly used “code of length N and cardinality T .”
The rate of a code of length N and cardinality T is, as usual, R =(logT )/N .Denoteby
N(T,w,r) the minimum possible length of a (w, r) cover-free code of a given length T .Inthe
present paper, we are interested in the asymptotic behavior of the rate
R(w, r) = lim sup
of such (optimal) codes.
A proof of an asymptotic lower bound for the rate of (w, r) cover-free codes, which uses the
random choice and expurgation technique, can be found in .
Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00098.
2003 MAIK “Nauka/Interperiodica”