3,925 research outputs found
A new structure for difference matrices over abelian -groups
A difference matrix over a group is a discrete structure that is intimately
related to many other combinatorial designs, including mutually orthogonal
Latin squares, orthogonal arrays, and transversal designs. Interest in
constructing difference matrices over -groups has been renewed by the recent
discovery that these matrices can be used to construct large linking systems of
difference sets, which in turn provide examples of systems of linked symmetric
designs and association schemes. We survey the main constructive and
nonexistence results for difference matrices, beginning with a classical
construction based on the properties of a finite field. We then introduce the
concept of a contracted difference matrix, which generates a much larger
difference matrix. We show that several of the main constructive results for
difference matrices over abelian -groups can be substantially simplified and
extended using contracted difference matrices. In particular, we obtain new
linking systems of difference sets of size in infinite families of abelian
-groups, whereas previously the largest known size was .Comment: 27 pages. Discussion of new reference [LT04
Properties and Construction of Polar Codes
Recently, Ar{\i}kan introduced the method of channel polarization on which
one can construct efficient capacity-achieving codes, called polar codes, for
any binary discrete memoryless channel. In the thesis, we show that decoding
algorithm of polar codes, called successive cancellation decoding, can be
regarded as belief propagation decoding, which has been used for decoding of
low-density parity-check codes, on a tree graph. On the basis of the
observation, we show an efficient construction method of polar codes using
density evolution, which has been used for evaluation of the error probability
of belief propagation decoding on a tree graph. We further show that channel
polarization phenomenon and polar codes can be generalized to non-binary
discrete memoryless channels. Asymptotic performances of non-binary polar
codes, which use non-binary matrices called the Reed-Solomon matrices, are
better than asymptotic performances of the best explicitly known binary polar
code. We also find that the Reed-Solomon matrices are considered to be natural
generalization of the original binary channel polarization introduced by
Ar{\i}kan.Comment: Master thesis. The supervisor is Toshiyuki Tanaka. 24 pages, 3
figure
Skew-symmetric distributions and Fisher information -- a tale of two densities
Skew-symmetric densities recently received much attention in the literature,
giving rise to increasingly general families of univariate and multivariate
skewed densities. Most of those families, however, suffer from the inferential
drawback of a potentially singular Fisher information in the vicinity of
symmetry. All existing results indicate that Gaussian densities (possibly after
restriction to some linear subspace) play a special and somewhat intriguing
role in that context. We dispel that widespread opinion by providing a full
characterization, in a general multivariate context, of the information
singularity phenomenon, highlighting its relation to a possible link between
symmetric kernels and skewing functions -- a link that can be interpreted as
the mismatch of two densities.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ346 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
List decoding of noisy Reed-Muller-like codes
First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are
two fundamental error-correcting codes which arise in communication as well as
in probabilistically-checkable proofs and learning. In this paper, we take the
first steps toward extending the quick randomized decoding tools of RM(1) into
the realm of quadratic binary and, equivalently, Z_4 codes. Our main
algorithmic result is an extension of the RM(1) techniques from Goldreich-Levin
and Kushilevitz-Mansour algorithms to the Hankel code, a code between RM(1) and
RM(2). That is, given signal s of length N, we find a list that is a superset
of all Hankel codewords phi with dot product to s at least (1/sqrt(k)) times
the norm of s, in time polynomial in k and log(N). We also give a new and
simple formulation of a known Kerdock code as a subcode of the Hankel code. As
a corollary, we can list-decode Kerdock, too. Also, we get a quick algorithm
for finding a sparse Kerdock approximation. That is, for k small compared with
1/sqrt{N} and for epsilon > 0, we find, in time polynomial in (k
log(N)/epsilon), a k-Kerdock-term approximation s~ to s with Euclidean error at
most the factor (1+epsilon+O(k^2/sqrt{N})) times that of the best such
approximation
Counting problems for special-orthogonal Anosov representations
For positive integers and let be the
projective indefinite special-orthogonal group of signature . We study
counting problems in the Riemannian symmetric space of and in the
pseudo-Riemannian hyperbolic space . Let be
a totally geodesic copy of . We look at the orbit of
under the action of a projective Anosov subgroup of . For certain
choices of such a geodesic copy we show that the number of points in this orbit
which are at distance at most from is finite and asymptotic to a purely
exponential function as goes to infinity. We provide an interpretation of
this result in , as the asymptotics of the amount of
space-like geodesic segments of maximum length in the orbit of a point.Comment: To appear in Ann. Inst. Fourier. 40 pages, 1 figur
Parrondo games as disordered systems
Parrondo's paradox refers to the counter-intuitive situation where a winning
strategy results from a suitable combination of losing ones. Simple stochastic
games exhibiting this paradox have been introduced around the turn of the
millennium. The common setting of these Parrondo games is that two rules,
and , are played at discrete time steps, following either a periodic pattern
or an aperiodic one, be it deterministic or random. These games can be mapped
onto 1D random walks. In capital-dependent games, the probabilities of moving
right or left depend on the walker's position modulo some integer . In
history-dependent games, each step is correlated with the previous ones. In
both cases the gain identifies with the velocity of the walker's ballistic
motion, which depends non-linearly on model parameters, allowing for the
possibility of Parrondo's paradox. Calculating the gain involves products of
non-commuting Markov matrices, which are somehow analogous to the transfer
matrices used in the physics of 1D disordered systems. Elaborating upon this
analogy, we study a paradigmatic Parrondo game of each class in the neutral
situation where each rule, when played alone, is fair. The main emphasis of
this systematic approach is on the dependence of the gain on the remaining
parameters and, above all, on the game, i.e., the rule pattern, be it periodic
or aperiodic, deterministic or random. One of the most original sides of this
work is the identification of weak-contrast regimes for capital-dependent and
history-dependent Parrondo games, and a detailed quantitative investigation of
the gain in the latter scaling regimes.Comment: 17 pages, 10 figures, 2 table
Open Problems on Central Simple Algebras
We provide a survey of past research and a list of open problems regarding
central simple algebras and the Brauer group over a field, intended both for
experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered,
compared to v
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