183,081 research outputs found
Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes
An optimal constant-composition or constant-weight code of weight has
linear size if and only if its distance is at least . When , the determination of the exact size of such a constant-composition or
constant-weight code is trivial, but the case of has been solved
previously only for binary and ternary constant-composition and constant-weight
codes, and for some sporadic instances.
This paper provides a construction for quasicyclic optimal
constant-composition and constant-weight codes of weight and distance
based on a new generalization of difference triangle sets. As a result,
the sizes of optimal constant-composition codes and optimal constant-weight
codes of weight and distance are determined for all such codes of
sufficiently large lengths. This solves an open problem of Etzion.
The sizes of optimal constant-composition codes of weight and distance
are also determined for all , except in two cases.Comment: 12 page
Approximate generalized Steiner systems and near-optimal constant weight codes
Constant weight codes (CWCs) and constant composition codes (CCCs) are two
important classes of codes that have been studied extensively in both
combinatorics and coding theory for nearly sixty years. In this paper we show
that for {\it all} fixed odd distances, there exist near-optimal CWCs and CCCs
asymptotically achieving the classic Johnson-type upper bounds.
Let denote the maximum size of -ary CWCs of length with
constant weight and minimum distance . One of our main results shows
that for {\it all} fixed and odd , one has
,
where . This implies the existence of near-optimal
generalized Steiner systems originally introduced by Etzion, and can be viewed
as a counterpart of a celebrated result of R\"odl on the existence of
near-optimal Steiner systems. Note that prior to our work, very little is known
about for . A similar result is proved for the maximum
size of CCCs.
We provide different proofs for our two main results, based on two
strengthenings of the well-known Frankl-R\"odl-Pippenger theorem on the
existence of near-optimal matchings in hypergraphs: the first proof follows by
Kahn's linear programming variation of the above theorem, and the second
follows by the recent independent work of Delcour-Postle, and
Glock-Joos-Kim-K\"uhn-Lichev on the existence of near-optimal matchings
avoiding certain forbidden configurations.
We also present several intriguing open questions for future research.Comment: 15 pages, introduction revise
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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