2,148 research outputs found
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
On the lengths of divisible codes
In this article, the effective lengths of all -divisible linear codes
over with a non-negative integer are determined. For that
purpose, the -adic expansion of an integer is introduced. It is
shown that there exists a -divisible -linear code of
effective length if and only if the leading coefficient of the
-adic expansion of is non-negative. Furthermore, the maximum weight
of a -divisible code of effective length is at most ,
where denotes the cross-sum of the -adic expansion of .
This result has applications in Galois geometries. A recent theorem of
N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a
corollary. Furthermore, we get an improvement of the Johnson bound for constant
dimension subspace codes.Comment: 17 pages, typos corrected; the paper was originally named "An
improvement of the Johnson bound for subspace codes
Covering of Subspaces by Subspaces
Lower and upper bounds on the size of a covering of subspaces in the
Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph
\cG_q(n,k), , are discussed. The problem is of interest from four
points of view: coding theory, combinatorial designs, -analogs, and
projective geometry. In particular we examine coverings based on lifted maximum
rank distance codes, combined with spreads and a recursive construction. New
constructions are given for with or . We discuss the density
for some of these coverings. Tables for the best known coverings, for and
, are presented. We present some questions concerning
possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352
Characterisations of elementary pseudo-caps and good eggs
In this note, we use the theory of Desarguesian spreads to investigate good
eggs. Thas showed that an egg in , odd, with two good
elements is elementary. By a short combinatorial argument, we show that a
similar statement holds for large pseudo-caps, in odd and even characteristic.
As a corollary, this improves and extends the result of Thas, Thas and Van
Maldeghem (2006) where one needs at least 4 good elements of an egg in even
characteristic to obtain the same conclusion. We rephrase this corollary to
obtain a characterisation of the generalised quadrangle of
Tits.
Lavrauw (2005) characterises elementary eggs in odd characteristic as those
good eggs containing a space that contains at least 5 elements of the egg, but
not the good element. We provide an adaptation of this characterisation for
weak eggs in odd and even characteristic. As a corollary, we obtain a direct
geometric proof for the theorem of Lavrauw
Identifying Infection Sources and Regions in Large Networks
Identifying the infection sources in a network, including the index cases
that introduce a contagious disease into a population network, the servers that
inject a computer virus into a computer network, or the individuals who started
a rumor in a social network, plays a critical role in limiting the damage
caused by the infection through timely quarantine of the sources. We consider
the problem of estimating the infection sources and the infection regions
(subsets of nodes infected by each source) in a network, based only on
knowledge of which nodes are infected and their connections, and when the
number of sources is unknown a priori. We derive estimators for the infection
sources and their infection regions based on approximations of the infection
sequences count. We prove that if there are at most two infection sources in a
geometric tree, our estimator identifies the true source or sources with
probability going to one as the number of infected nodes increases. When there
are more than two infection sources, and when the maximum possible number of
infection sources is known, we propose an algorithm with quadratic complexity
to estimate the actual number and identities of the infection sources.
Simulations on various kinds of networks, including tree networks, small-world
networks and real world power grid networks, and tests on two real data sets
are provided to verify the performance of our estimators
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