5,952 research outputs found
New Families of -ary Sequences of Period With Low Maximum Correlation Magnitude
Let be an odd prime such that and be an
odd integer. In this paper, two new families of -ary sequences of period are constructed by two decimated -ary m-sequences
and , where and . The upper bound on the
magnitude of correlation values of two sequences in the family is derived using
Weil bound. Their upper bound is derived as and the family size is 4N, which is four
times the period of the sequence.Comment: 9 page, no figure
Kloosterman paths and the shape of exponential sums
We consider the distribution of the polygonal paths joining partial sums of
classical Kloosterman sums, as their parameter varies modulo a prime tending to
infinity. Using independence of Kloosterman sheaves, we prove convergence in
the sense of finite distributions to a specific random Fourier series. We also
consider Birch sums, for which we can establish convergence in law in the space
of continuous functions. We then derive some applications.Comment: 27 pages, 3 figure
Infinite families of -designs from a class of cyclic codes with two non-zeros
Combinatorial -designs have wide applications in coding theory,
cryptography, communications and statistics. It is well known that the supports
of all codewords with a fixed weight in a code may give a -design. In this
paper, we first determine the weight distribution of a class of linear codes
derived from the dual of extended cyclic code with two non-zeros. We then
obtain infinite families of -designs and explicitly compute their parameters
from the supports of all the codewords with a fixed weight in the codes. By
simple counting argument, we obtain exponentially many -designs.Comment: arXiv admin note: substantial text overlap with arXiv:1903.0745
Selfdecomposability and selfsimilarity: a concise primer
We summarize the relations among three classes of laws: infinitely divisible,
selfdecomposable and stable. First we look at them as the solutions of the
Central Limit Problem; then their role is scrutinized in relation to the Levy
and the additive processes with an emphasis on stationarity and selfsimilarity.
Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy noises and
their selfdecomposable stationary distributions, and we end with a few
particular examples.Comment: 24 pages, 3 figures; corrected misprint in the title; redactional
modifications required by the referee; added references from [16] to [28];.
Accepted and in press on Physica
Infinite families of -designs from two classes of linear codes
The interplay between coding theory and -designs has attracted a lot of
attention for both directions. It is well known that the supports of all
codewords with a fixed weight in a code may hold a -design. In this paper,
by determining the weight distributions of two classes of linear codes, we
derive infinite families of -designs from the supports of codewords with a
fixed weight in these codes, and explicitly obtain their parameters
Random matrix theory, the exceptional Lie groups, and L-functions
There has recently been interest in relating properties of matrices drawn at
random from the classical compact groups to statistical characteristics of
number-theoretical L-functions. One example is the relationship conjectured to
hold between the value distributions of the characteristic polynomials of such
matrices and value distributions within families of L-functions. These
connections are here extended to non-classical groups. We focus on an explicit
example: the exceptional Lie group G_2. The value distributions for
characteristic polynomials associated with the 7- and 14-dimensional
representations of G_2, defined with respect to the uniform invariant (Haar)
measure, are calculated using two of the Macdonald constant term identities. A
one parameter family of L-functions over a finite field is described whose
value distribution in the limit as the size of the finite field grows is
related to that of the characteristic polynomials associated with the
7-dimensional representation of G_2. The random matrix calculations extend to
all exceptional Lie groupsComment: 14 page
Gaussian distribution of short sums of trace functions over finite fields
We show that under certain general conditions, short sums of -adic
trace functions over finite fields follow a normal distribution asymptotically
when the origin varies, generalizing results of Erd\H{o}s-Davenport,
Mak-Zaharescu and Lamzouri. In particular, this applies to exponential sums
arising from Fourier transforms such as Kloosterman sums or Birch sums, as we
can deduce from the works of Katz. By approximating the moments of traces of
random matrices in monodromy groups, a quantitative version can be given as in
Lamzouri's article, exhibiting a different phenomenon than the averaging from
the central limit theorem.Comment: 42 page
Exponential sums and finite field -hypergeometric functions
We define finite field -hypergeometric functions and show that they are
Fourier expansions of families of exponential sums on the torus. For an
appropriate choice of , our finite field -hypergeometric function can be
specialized to the finite field -hypergeometric function defined
by McCarthy.Comment: 4 page
An Exponential Sum and Higher-Codimensional Subvarieties of Projective Spaces over Finite Fields
A general method to express in terms of Gauss sums the number of rational
points of subschemes of projective schemes over finite fields is applied to the
image of the triple embedding . As a
consequence, we obtain a non-trivial description of the value of a
Kloosterman-sum-like exponential sum.Comment: 12 pages; to appear in Hiroshima Mathematical Journa
Hasse invariants and mod solutions of -hypergeometric systems
Igusa noted that the Hasse invariant of the Legendre family of elliptic
curves over a finite field of odd characteristic is a solution mod of a
Gaussian hypergeometric equation. We show that any family of exponential sums
over a finite field has a Hasse invariant which is a sum of products of mod
solutions of -hypergeometric systems.Comment: 22 page
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