4,959 research outputs found
Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three
The concept of group divisible codes, a generalization of group divisible
designs with constant block size, is introduced in this paper. This new class
of codes is shown to be useful in recursive constructions for constant-weight
and constant-composition codes. Large classes of group divisible codes are
constructed which enabled the determination of the sizes of optimal
constant-composition codes of weight three (and specified distance), leaving
only four cases undetermined. Previously, the sizes of constant-composition
codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table
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
Bounds for DNA codes with constant GC-content
We derive theoretical upper and lower bounds on the maximum size of DNA codes
of length n with constant GC-content w and minimum Hamming distance d, both
with and without the additional constraint that the minimum Hamming distance
between any codeword and the reverse-complement of any codeword be at least d.
We also explicitly construct codes that are larger than the best
previously-published codes for many choices of the parameters n, d and w.Comment: 13 pages, no figures; a few references added and typos correcte
The PBD-Closure of Constant-Composition Codes
We show an interesting PBD-closure result for the set of lengths of
constant-composition codes whose distance and size meet certain conditions. A
consequence of this PBD-closure result is that the size of optimal
constant-composition codes can be determined for infinite families of parameter
sets from just a single example of an optimal code. As an application, the size
of several infinite families of optimal constant-composition codes are derived.
In particular, the problem of determining the size of optimal
constant-composition codes having distance four and weight three is solved for
all lengths sufficiently large. This problem was previously unresolved for odd
lengths, except for lengths seven and eleven.Comment: 8 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
Importance of Symbol Equity in Coded Modulation for Power Line Communications
The use of multiple frequency shift keying modulation with permutation codes
addresses the problem of permanent narrowband noise disturbance in a power line
communications system. In this paper, we extend this coded modulation scheme
based on permutation codes to general codes and introduce an additional new
parameter that more precisely captures a code's performance against permanent
narrowband noise. As a result, we define a new class of codes, namely,
equitable symbol weight codes, which are optimal with respect to this measure
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