2,351 research outputs found
Sphere packing bounds via spherical codes
The sphere packing problem asks for the greatest density of a packing of
congruent balls in Euclidean space. The current best upper bound in all
sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We
revisit their argument and improve their bound by a constant factor using a
simple geometric argument, and we extend the argument to packings in hyperbolic
space, for which it gives an exponential improvement over the previously known
bounds. Additionally, we show that the Cohn-Elkies linear programming bound is
always at least as strong as the Kabatiansky-Levenshtein bound; this result is
analogous to Rodemich's theorem in coding theory. Finally, we develop
hyperbolic linear programming bounds and prove the analogue of Rodemich's
theorem there as well.Comment: 30 pages, 2 figure
The DMT classification of real and quaternionic lattice codes
In this paper we consider space-time codes where the code-words are
restricted to either real or quaternion matrices. We prove two separate
diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and
provide a criterion for a lattice code to achieve these upper bounds. We also
point out that lattice codes based on Q-central division algebras satisfy this
optimality criterion. As a corollary this result provides a DMT classification
for all Q-central division algebra codes that are based on standard embeddings.Comment: 6 pages, 1 figure. Conference paper submitted to the International
Symposium on Information Theory 201
Distributed Structure: Joint Expurgation for the Multiple-Access Channel
In this work we show how an improved lower bound to the error exponent of the
memoryless multiple-access (MAC) channel is attained via the use of linear
codes, thus demonstrating that structure can be beneficial even in cases where
there is no capacity gain. We show that if the MAC channel is modulo-additive,
then any error probability, and hence any error exponent, achievable by a
linear code for the corresponding single-user channel, is also achievable for
the MAC channel. Specifically, for an alphabet of prime cardinality, where
linear codes achieve the best known exponents in the single-user setting and
the optimal exponent above the critical rate, this performance carries over to
the MAC setting. At least at low rates, where expurgation is needed, our
approach strictly improves performance over previous results, where expurgation
was used at most for one of the users. Even when the MAC channel is not
additive, it may be transformed into such a channel. While the transformation
is lossy, we show that the distributed structure gain in some "nearly additive"
cases outweighs the loss, and thus the error exponent can improve upon the best
known error exponent for these cases as well. Finally we apply a similar
approach to the Gaussian MAC channel. We obtain an improvement over the best
known achievable exponent, given by Gallager, for certain rate pairs, using
lattice codes which satisfy a nesting condition.Comment: Submitted to the IEEE Trans. Info. Theor
Local Testing for Membership in Lattices
Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)
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