28,561 research outputs found
A General Upper Bound on the Size of Constant-Weight Conflict-Avoiding Codes
Conflict-avoiding codes are used in the multiple-access collision channel
without feedback. The number of codewords in a conflict-avoiding code is the
number of potential users that can be supported in the system. In this paper, a
new upper bound on the size of conflict-avoiding codes is proved. This upper
bound is general in the sense that it is applicable to all code lengths and all
Hamming weights. Several existing constructions for conflict-avoiding codes,
which are known to be optimal for Hamming weights equal to four and five, are
shown to be optimal for all Hamming weights in general.Comment: 10 pages, 1 figur
Determination of the exponent gamma for SAWs on the two-dimensional Manhattan lattice
We present a high-statistics Monte Carlo determination of the exponent gamma
for self-avoiding walks on a Manhattan lattice in two dimensions. A
conservative estimate is \gamma \gtapprox 1.3425(3), in agreement with the
universal value 43/32 on regular lattices, but in conflict with predictions
from conformal field theory and with a recent estimate from exact enumerations.
We find strong corrections to scaling that seem to indicate the presence of a
non-analytic exponent Delta < 1. If we assume Delta = 11/16 we find gamma =
1.3436(3), where the error is purely statistical.Comment: 24 pages, LaTeX2e, 4 figure
Partition Information and its Transmission over Boolean Multi-Access Channels
In this paper, we propose a novel partition reservation system to study the
partition information and its transmission over a noise-free Boolean
multi-access channel. The objective of transmission is not message restoration,
but to partition active users into distinct groups so that they can,
subsequently, transmit their messages without collision. We first calculate (by
mutual information) the amount of information needed for the partitioning
without channel effects, and then propose two different coding schemes to
obtain achievable transmission rates over the channel. The first one is the
brute force method, where the codebook design is based on centralized source
coding; the second method uses random coding where the codebook is generated
randomly and optimal Bayesian decoding is employed to reconstruct the
partition. Both methods shed light on the internal structure of the partition
problem. A novel hypergraph formulation is proposed for the random coding
scheme, which intuitively describes the information in terms of a strong
coloring of a hypergraph induced by a sequence of channel operations and
interactions between active users. An extended Fibonacci structure is found for
a simple, but non-trivial, case with two active users. A comparison between
these methods and group testing is conducted to demonstrate the uniqueness of
our problem.Comment: Submitted to IEEE Transactions on Information Theory, major revisio
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
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