502 research outputs found
On the upper bound of the size of the r-cover-free families
Let T (r; n) denote the maximum number of subsets of an n-set satisfying the condition in the title. It is proved in a purely combinatorial way, that for n sufficiently large log 2 T (r; n) n 8 \Delta log 2 r r 2 holds. 1. Introduction The notion of the r-cover-free families was introduced by Kautz and Singleton in 1964 [17]. They initiated investigating binary codes with the property that the disjunction of any r (r 2) codewords are distinct (UD r codes). This led them to studying the binary codes with the property that none of the codewords is covered by the disjunction of r others (Superimposed codes, ZFD r codes; P. Erdos, P. Frankl and Z. Furedi called the correspondig set system r-cover-free in [7]). Since that many results have been proved about the maximum size of these codes. Various authors studied these problems basically from three different points of view, and these three lines of investigations were almost independent of each other. This is why many results were ..
Symmetric Disjunctive List-Decoding Codes
A binary code is said to be a disjunctive list-decoding -code (LD
-code), , , if the code is identified by the incidence
matrix of a family of finite sets in which the union (or disjunctive sum) of
any sets can cover not more than other sets of the family. In this
paper, we consider a similar class of binary codes which are based on a {\em
symmetric disjunctive sum} (SDS) of binary symbols. By definition, the
symmetric disjunctive sum (SDS) takes values from the ternary alphabet , where the symbol~ denotes "erasure". Namely: SDS is equal to ()
if all its binary symbols are equal to (), otherwise SDS is equal
to~. List decoding codes for symmetric disjunctive sum are said to be {\em
symmetric disjunctive list-decoding -codes} (SLD -codes). In the
given paper, we remind some applications of SLD -codes which motivate the
concept of symmetric disjunctive sum. We refine the known relations between
parameters of LD -codes and SLD -codes. For the ensemble of binary
constant-weight codes we develop a random coding method to obtain lower bounds
on the rate of these codes. Our lower bounds improve the known random coding
bounds obtained up to now using the ensemble with independent symbols of
codewords.Comment: 18 pages, 1 figure, 1 table, conference pape
Construction of Almost Disjunct Matrices for Group Testing
In a \emph{group testing} scheme, a set of tests is designed to identify a
small number of defective items among a large set (of size ) of items.
In the non-adaptive scenario the set of tests has to be designed in one-shot.
In this setting, designing a testing scheme is equivalent to the construction
of a \emph{disjunct matrix}, an matrix where the union of supports
of any columns does not contain the support of any other column. In
principle, one wants to have such a matrix with minimum possible number of
rows (tests). One of the main ways of constructing disjunct matrices relies on
\emph{constant weight error-correcting codes} and their \emph{minimum
distance}. In this paper, we consider a relaxed definition of a disjunct matrix
known as \emph{almost disjunct matrix}. This concept is also studied under the
name of \emph{weakly separated design} in the literature. The relaxed
definition allows one to come up with group testing schemes where a
close-to-one fraction of all possible sets of defective items are identifiable.
Our main contribution is twofold. First, we go beyond the minimum distance
analysis and connect the \emph{average distance} of a constant weight code to
the parameters of an almost disjunct matrix constructed from it. Our second
contribution is to explicitly construct almost disjunct matrices based on our
average distance analysis, that have much smaller number of rows than any
previous explicit construction of disjunct matrices. The parameters of our
construction can be varied to cover a large range of relations for and .Comment: 15 Page
Group testing problems in experimental molecular biology
In group testing, the task is to determine the distinguished members of a set
of objects L by asking subset queries of the form ``does the subset Q of L
contain a distinguished object?'' The primary biological application of group
testing is for screening libraries of clones with hybridization probes. This is
a crucial step in constructing physical maps and for finding genes. Group
testing has also been considered for sequencing by hybridization. Another
important application includes screening libraries of reagents for useful
chemically active zones. This preliminary report discusses some of the
constrained group testing problems which arise in biology.Comment: 7 page
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