35,314 research outputs found
New constructions for covering designs
A {\em covering design}, or {\em covering}, is a family of
-subsets, called blocks, chosen from a -set, such that each -subset is
contained in at least one of the blocks. The number of blocks is the covering's
{\em size}, and the minimum size of such a covering is denoted by .
This paper gives three new methods for constructing good coverings: a greedy
algorithm similar to Conway and Sloane's algorithm for lexicographic
codes~\cite{lex}, and two methods that synthesize new coverings from
preexisting ones. Using these new methods, together with results in the
literature, we build tables of upper bounds on for , , and .
Partial Covering Arrays: Algorithms and Asymptotics
A covering array is an array with entries
in , for which every subarray contains each
-tuple of among its rows. Covering arrays find
application in interaction testing, including software and hardware testing,
advanced materials development, and biological systems. A central question is
to determine or bound , the minimum number of rows of
a . The well known bound
is not too far from being
asymptotically optimal. Sensible relaxations of the covering requirement arise
when (1) the set need only be contained among the rows
of at least of the subarrays and (2) the
rows of every subarray need only contain a (large) subset of . In this paper, using probabilistic methods, significant
improvements on the covering array upper bound are established for both
relaxations, and for the conjunction of the two. In each case, a randomized
algorithm constructs such arrays in expected polynomial time
On Asymmetric Coverings and Covering Numbers
An asymmetric covering D(n,R) is a collection of special subsets S of an
n-set such that every subset T of the n-set is contained in at least one
special S with |S| - |T| <= R. In this paper we compute the smallest size of
any D(n,1) for n <= 8. We also investigate ``continuous'' and ``banded''
versions of the problem. The latter involves the classical covering numbers
C(n,k,k-1), and we determine the following new values: C(10,5,4) = 51,
C(11,7,6,) =84, C(12,8,7) = 126, C(13,9,8)= 185 and C(14,10,9) = 259. We also
find the number of nonisomorphic minimal covering designs in several cases.Comment: 11 page
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