5,846 research outputs found
New Bounds for Permutation Codes in Ulam Metric
New bounds on the cardinality of permutation codes equipped with the Ulam
distance are presented. First, an integer-programming upper bound is derived,
which improves on the Singleton-type upper bound in the literature for some
lengths. Second, several probabilistic lower bounds are developed, which
improve on the known lower bounds for large minimum distances. The results of a
computer search for permutation codes are also presented.Comment: To be presented at ISIT 2015, 5 page
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Semidefinite programming, harmonic analysis and coding theory
These lecture notes where presented as a course of the CIMPA summer school in
Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics.
This version is an update June 2010
Time lower bounds for nonadaptive turnstile streaming algorithms
We say a turnstile streaming algorithm is "non-adaptive" if, during updates,
the memory cells written and read depend only on the index being updated and
random coins tossed at the beginning of the stream (and not on the memory
contents of the algorithm). Memory cells read during queries may be decided
upon adaptively. All known turnstile streaming algorithms in the literature are
non-adaptive.
We prove the first non-trivial update time lower bounds for both randomized
and deterministic turnstile streaming algorithms, which hold when the
algorithms are non-adaptive. While there has been abundant success in proving
space lower bounds, there have been no non-trivial update time lower bounds in
the turnstile model. Our lower bounds hold against classically studied problems
such as heavy hitters, point query, entropy estimation, and moment estimation.
In some cases of deterministic algorithms, our lower bounds nearly match known
upper bounds
Algorithmic patterns for -matrices on many-core processors
In this work, we consider the reformulation of hierarchical ()
matrix algorithms for many-core processors with a model implementation on
graphics processing units (GPUs). matrices approximate specific
dense matrices, e.g., from discretized integral equations or kernel ridge
regression, leading to log-linear time complexity in dense matrix-vector
products. The parallelization of matrix operations on many-core
processors is difficult due to the complex nature of the underlying algorithms.
While previous algorithmic advances for many-core hardware focused on
accelerating existing matrix CPU implementations by many-core
processors, we here aim at totally relying on that processor type. As main
contribution, we introduce the necessary parallel algorithmic patterns allowing
to map the full matrix construction and the fast matrix-vector
product to many-core hardware. Here, crucial ingredients are space filling
curves, parallel tree traversal and batching of linear algebra operations. The
resulting model GPU implementation hmglib is the, to the best of the authors
knowledge, first entirely GPU-based Open Source matrix library of
this kind. We conclude this work by an in-depth performance analysis and a
comparative performance study against a standard matrix library,
highlighting profound speedups of our many-core parallel approach
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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