12,056 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
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
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