39,687 research outputs found
Perfect forms over totally real number fields
A rational positive-definite quadratic form is perfect if it can be
reconstructed from the knowledge of its minimal nonzero value m and the finite
set of integral vectors v such that f(v) = m. This concept was introduced by
Voronoi and later generalized by Koecher to arbitrary number fields. One knows
that up to a natural "change of variables'' equivalence, there are only
finitely many perfect forms, and given an initial perfect form one knows how to
explicitly compute all perfect forms up to equivalence. In this paper we
investigate perfect forms over totally real number fields. Our main result
explains how to find an initial perfect form for any such field. We also
compute the inequivalent binary perfect forms over real quadratic fields
Q(\sqrt{d}) with d \leq 66.Comment: 11 pages, 2 figures, 1 tabl
Nonnegative polynomials and their Carath\'eodory number
In 1888 Hilbert showed that every nonnegative homogeneous polynomial with
real coefficients of degree in variables is a sum of squares if and
only if (quadratic forms), (binary forms) or (ternary
quartics). In these cases, it is interesting to compute canonical expressions
for these decompositions. Starting from Carath\'eodory's Theorem, we compute
the Carath\'eodory number of Hilbert cones of nonnegative quadratic and binary
forms.Comment: 9 pages. Discrete & Computational Geometry (2014
Linear Codes from Some 2-Designs
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -design as a generator
matrix over \gf(q) of the code. This approach has been extensively
investigated in the literature. In this paper, a different method of
constructing linear codes using specific classes of -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -designs associated to semibent functions.
Two families of the codes obtained in this paper are optimal. The linear codes
presented in this paper have applications in secret sharing and authentication
schemes, in addition to their applications in consumer electronics,
communication and data storage systems. A coding-theory approach to the
characterisation of highly nonlinear Boolean functions is presented
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