42,535 research outputs found
Non-asymptotic Upper Bounds for Deletion Correcting Codes
Explicit non-asymptotic upper bounds on the sizes of multiple-deletion
correcting codes are presented. In particular, the largest single-deletion
correcting code for -ary alphabet and string length is shown to be of
size at most . An improved bound on the asymptotic
rate function is obtained as a corollary. Upper bounds are also derived on
sizes of codes for a constrained source that does not necessarily comprise of
all strings of a particular length, and this idea is demonstrated by
application to sets of run-length limited strings.
The problem of finding the largest deletion correcting code is modeled as a
matching problem on a hypergraph. This problem is formulated as an integer
linear program. The upper bound is obtained by the construction of a feasible
point for the dual of the linear programming relaxation of this integer linear
program.
The non-asymptotic bounds derived imply the known asymptotic bounds of
Levenshtein and Tenengolts and improve on known non-asymptotic bounds.
Numerical results support the conjecture that in the binary case, the
Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure
Partial ovoids and partial spreads in symplectic and orthogonal polar spaces
We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)]
New upper bounds for spherical codes and packings
We improve the previously best known upper bounds on the sizes of
-spherical codes for every at
least by a factor of , in sufficiently high dimensions. Furthermore,
for sphere packing densities in dimensions we have an improvement
at least by a factor of . Our method also breaks many
non-numerical sphere packing density bounds in small dimensions. Apart from
Cohn and Zhao's \cite{CohnZhao} improvement on the geometric average of
Levenshtein's bound \cite{Leven79} over all sufficiently high dimensions by a
factor of our work is the first improvement for each dimension since
the work of Kabatyanskii and Levenshtein \cite{KL} and its later improvement by
Levenshtein \cite{Leven79}. Moreover, we generalize Levenshtein's optimal
polynomials and provide explicit formulae for them that may be of independent
interest. For we construct a test function for Delsarte's
linear programing problem for -spherical codes with exponentially
improved factor in dimension compared to previous test functions.Comment: Comments are welcome
Tables of subspace codes
One of the main problems of subspace coding asks for the maximum possible
cardinality of a subspace code with minimum distance at least over
, where the dimensions of the codewords, which are vector
spaces, are contained in . In the special case of
one speaks of constant dimension codes. Since this (still) emerging
field is very prosperous on the one hand side and there are a lot of
connections to classical objects from Galois geometry it is a bit difficult to
keep or to obtain an overview about the current state of knowledge. To this end
we have implemented an on-line database of the (at least to us) known results
at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated
technical report is to provide a user guide how this technical tool can be used
in research projects and to describe the so far implemented theoretic and
algorithmic knowledge.Comment: 44 pages, 6 tables, 7 screenshot
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
Estimates on the Size of Symbol Weight Codes
The study of codes for powerlines communication has garnered much interest
over the past decade. Various types of codes such as permutation codes,
frequency permutation arrays, and constant composition codes have been proposed
over the years. In this work we study a type of code called the bounded symbol
weight codes which was first introduced by Versfeld et al. in 2005, and a
related family of codes that we term constant symbol weight codes. We provide
new upper and lower bounds on the size of bounded symbol weight and constant
symbol weight codes. We also give direct and recursive constructions of codes
for certain parameters.Comment: 14 pages, 4 figure
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