2,186 research outputs found
R.A.Fisher, design theory, and the Indian connection
Design Theory, a branch of mathematics, was born out of the experimental
statistics research of the population geneticist R. A. Fisher and of Indian
mathematical statisticians in the 1930s. The field combines elements of
combinatorics, finite projective geometries, Latin squares, and a variety of
further mathematical structures, brought together in surprising ways. This
essay will present these structures and ideas as well as how the field came
together, in itself an interesting story.Comment: 11 pages, 3 figure
Efficient Two-Stage Group Testing Algorithms for Genetic Screening
Efficient two-stage group testing algorithms that are particularly suited for
rapid and less-expensive DNA library screening and other large scale biological
group testing efforts are investigated in this paper. The main focus is on
novel combinatorial constructions in order to minimize the number of individual
tests at the second stage of a two-stage disjunctive testing procedure.
Building on recent work by Levenshtein (2003) and Tonchev (2008), several new
infinite classes of such combinatorial designs are presented.Comment: 14 pages; to appear in "Algorithmica". Part of this work has been
presented at the ICALP 2011 Group Testing Workshop; arXiv:1106.368
Probabilistic Existence of Large Sets of Designs
A new probabilistic technique for establishing the existence of certain
regular combinatorial structures has been recentlyintroduced by Kuperberg,
Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under
certain conditions, a randomly chosen structure has the required properties of
a - combinatorial design with tiny, yet positive,
probability.
The proof method of KLP is adapted to show the existence of large sets of
designs and similar combinatorial structures as follows. We modify the random
choice and the analysis to show that, under the same conditions, not only does
a - design exist but, in fact, with positive probability
there exists a large set of such designs -- that is, a partition of the set of
-subsets of into -designs - designs.
Specifically, using the probabilistic approach derived herein, we prove that
for all sufficiently large , large sets of - designs exist
whenever and the necessary divisibility conditions are satisfied.
This resolves the existence conjecture for large sets of designs for all .Comment: 20 page
A tripling construction for overlarge sets of KTS
AbstractAn overlarge set of KTS(v), denoted by OLKTS(v), is a collection {(X∖{x},Bx):x∈X}, where X is a (v+1)-set, each (X∖{x},Bx) is a KTS(v) and {Bx:x∈X} forms a partition of all triples on X. In this paper, we give a tripling construction for overlarge sets of KTS. Our main result is that: If there exists an OLKTS(v) with a special property, then there exists an OLKTS(3v). It is obtained that there exists an OLKTS(3m(2u+1)) for u=22n−1−1 or u=qn, where prime power q≡7 (mod 12) and m≥0,n≥1
Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three
The concept of group divisible codes, a generalization of group divisible
designs with constant block size, is introduced in this paper. This new class
of codes is shown to be useful in recursive constructions for constant-weight
and constant-composition codes. Large classes of group divisible codes are
constructed which enabled the determination of the sizes of optimal
constant-composition codes of weight three (and specified distance), leaving
only four cases undetermined. Previously, the sizes of constant-composition
codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table
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