395,100 research outputs found
Almost Cover-Free Codes and Designs
An -subset of codewords of a binary code is said to be an {\em
-bad} in if the code contains a subset of other
codewords such that the conjunction of the codewords is covered by the
disjunctive sum of the codewords. Otherwise, the -subset of codewords of
is said to be an {\em -good} in~.mA binary code is said to
be a cover-free -code if the code does not contain -bad
subsets. In this paper, we introduce a natural {\em probabilistic}
generalization of cover-free -codes, namely: a binary code is said to
be an almost cover-free -code if {\em almost all} -subsets of its
codewords are -good. We discuss the concept of almost cover-free
-codes arising in combinatorial group testing problems connected with
the nonadaptive search of defective supersets (complexes). We develop a random
coding method based on the ensemble of binary constant weight codes to obtain
lower bounds on the capacity of such codes.Comment: 18 pages, conference pape
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 ..
Uniform hypergraphs containing no grids
A hypergraph is called an rĂr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that AiâŠAj=BiâŠBj=Ď for 1â¤i<jâ¤r and {pipe}AiâŠBj{pipe}=1 for 1â¤i, jâ¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1âŠC2{pipe}={pipe}C2âŠC3{pipe}={pipe}C3âŠC1{pipe}=1, C1âŠC2â C1âŠC3. A hypergraph is linear, if {pipe}EâŠF{pipe}â¤1 holds for every pair of edges Eâ F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For râĽ. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Š 2013 Elsevier Ltd
2-cancellative hypergraphs and codes
A family of sets F (and the corresponding family of 0-1 vectors) is called
t-cancellative if for all distict t+2 members A_1,... A_t and B,C from F the
union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let
c(n,t) be the size of the largest t-cancellative family on n elements, and let
c_k(n,t) denote the largest k-uniform family. We significantly improve the
previous upper bounds, e.g., we show c(n,2) n_0). Using an
algebraic construction we show that the order of magnitude of c_{2k}(n,2) is
n^k for each k (when n goes to infinity).Comment: 20 page
Modeled flux and polarisation signals of horizontally inhomogeneous exoplanets, applied to Earth--like planets
We present modeled flux and linear polarisation signals of starlight that is
reflected by spatially unresolved, horizontally inhomogeneous planets and
discuss the effects of including horizontal inhomogeneities on the flux and
polarisation signals of Earth-like exoplanets. Our code is based on an
efficient adding--doubling algorithm, which fully includes multiple scattering
by gases and aerosol/cloud particles. We divide a model planet into pixels that
are small enough for the local properties of the atmosphere and surface (if
present) to be horizontally homogeneous. Given a planetary phase angle, we sum
up the reflected total and linearly polarised fluxes across the illuminated and
visible part of the planetary disk, taking care to properly rotate the
polarized flux vectors towards the same reference plane. We compared flux and
polarisation signals of simple horizontally inhomogeneous model planets against
results of the weighted sum approximation, in which signals of horizontally
homogeneous planets are combined. Apart from cases in which the planet has only
a minor inhomogeneity, the signals differ significantly. In particular, the
shape of the polarisation phase function appears to be sensitive to the
horizontal inhomogeneities. The same holds true for Earth-like model planets
with patchy clouds above an ocean and a sandy continent. Our simulations
clearly show that horizontal inhomogeneities leave different traces in flux and
polarisation signals. Combining flux with polarisation measurements would help
retrieving the atmospheric and surface patterns on a planet
Constraining the Number of Positive Responses in Adaptive, Non-Adaptive, and Two-Stage Group Testing
Group testing is a well known search problem that consists in detecting the
defective members of a set of objects O by performing tests on properly chosen
subsets (pools) of the given set O. In classical group testing the goal is to
find all defectives by using as few tests as possible. We consider a variant of
classical group testing in which one is concerned not only with minimizing the
total number of tests but aims also at reducing the number of tests involving
defective elements. The rationale behind this search model is that in many
practical applications the devices used for the tests are subject to
deterioration due to exposure to or interaction with the defective elements. In
this paper we consider adaptive, non-adaptive and two-stage group testing. For
all three considered scenarios, we derive upper and lower bounds on the number
of "yes" responses that must be admitted by any strategy performing at most a
certain number t of tests. In particular, for the adaptive case we provide an
algorithm that uses a number of "yes" responses that exceeds the given lower
bound by a small constant. Interestingly, this bound can be asymptotically
attained also by our two-stage algorithm, which is a phenomenon analogous to
the one occurring in classical group testing. For the non-adaptive scenario we
give almost matching upper and lower bounds on the number of "yes" responses.
In particular, we give two constructions both achieving the same asymptotic
bound. An interesting feature of one of these constructions is that it is an
explicit construction. The bounds for the non-adaptive and the two-stage cases
follow from the bounds on the optimal sizes of new variants of d-cover free
families and (p,d)-cover free families introduced in this paper, which we
believe may be of interest also in other contexts
On Multistage Learning a Hidden Hypergraph
Learning a hidden hypergraph is a natural generalization of the classical
group testing problem that consists in detecting unknown hypergraph
by carrying out edge-detecting tests. In the given paper we
focus our attention only on a specific family of localized
hypergraphs for which the total number of vertices , the number of
edges , , and the cardinality of any edge ,
. Our goal is to identify all edges of by
using the minimal number of tests. We develop an adaptive algorithm that
matches the information theory bound, i.e., the total number of tests of the
algorithm in the worst case is at most . We also discuss
a probabilistic generalization of the problem.Comment: 5 pages, IEEE conferenc
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