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Some New Bounds For Cover-Free Families Through Biclique Cover
An cover-free family is a family of subsets of a finite set
such that the intersection of any members of the family contains at least
elements that are not in the union of any other members. The minimum
number of elements for which there exists an with blocks is
denoted by .
In this paper, we show that the value of is equal to the
-biclique covering number of the bipartite graph whose vertices
are all - and -subsets of a -element set, where a -subset is
adjacent to an -subset if their intersection is empty. Next, we introduce
some new bounds for . For instance, we show that for
and
where is a constant satisfies the
well-known bound . Also, we
determine the exact value of for some values of . Finally, we
show that whenever there exists a Hadamard matrix of
order 4d
On Bounded Weight Codes
The maximum size of a binary code is studied as a function of its length N,
minimum distance D, and minimum codeword weight W. This function B(N,D,W) is
first characterized in terms of its exponential growth rate in the limit as N
tends to infinity for fixed d=D/N and w=W/N. The exponential growth rate of
B(N,D,W) is shown to be equal to the exponential growth rate of A(N,D) for w <=
1/2, and equal to the exponential growth rate of A(N,D,W) for 1/2< w <= 1.
Second, analytic and numerical upper bounds on B(N,D,W) are derived using the
semidefinite programming (SDP) method. These bounds yield a non-asymptotic
improvement of the second Johnson bound and are tight for certain values of the
parameters
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