307 research outputs found
A note on the generalized Hamming weights of Reed-Muller codes
In this note, we give a very simple description of the generalized Hamming
weights of Reed--Muller codes. For this purpose, we generalize the well-known
Macaulay representation of a nonnegative integer and state some of its basic
properties.Comment: 8 page
Explicit MDS Codes with Complementary Duals
In 1964, Massey introduced a class of codes with complementary duals which
are called Linear Complimentary Dual (LCD for short) codes. He showed that LCD
codes have applications in communication system, side-channel attack (SCA) and
so on. LCD codes have been extensively studied in literature. On the other
hand, MDS codes form an optimal family of classical codes which have wide
applications in both theory and practice. The main purpose of this paper is to
give an explicit construction of several classes of LCD MDS codes, using tools
from algebraic function fields. We exemplify this construction and obtain
several classes of explicit LCD MDS codes for the odd characteristic case
On the Deuring Polynomial for Drinfeld Modules in Legendre Form
We study a family of -Drinfeld modules,
which is a natural analog of Legendre elliptic curves. We then find a
surprising recurrence giving the corresponding Deuring polynomial
characterising supersingular Legendre Drinfeld modules
in characteristic .Comment: This article supersedes arXiv:1110.607
A new family of maximal curves
In this article we construct for any prime power and odd , a new
-maximal curve . Like the Garcia--G\"
uneri--Stichtenoth maximal curves, our curves generalize the
Giulietti--Korchm\'aros maximal curve, though in a different way. We compute
the full automorphism group of , yielding that it has precisely
automorphisms. Further, we show that unless , the curve
is not a Galois subcover of the Hermitian curve. Finally, we
find new values of the genus spectrum of -maximal curves,
by considering some Galois subcovers of
Maximum number of points on intersection of a cubic surface and a non-degenerate Hermitian surface
In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on
the intersection of a surface of degree and a non-degenerate Hermitian
surface in \PP^3(\Fqt). The conjecture was proven to be true by Edoukou in
the case when . In this paper, we prove that the conjecture is true for
and . We further determine the second highest number of rational
points on the intersection of a cubic surface and a non-degenerate Hermitian
surface. Finally, we classify all the cubic surfaces that admit the highest and
second highest number of points in common with a non-degenerate Hermitian
surface. This classifications disproves one of the conjectures proposed by
Edoukou, Ling and Xing
Generalized Hamming weights of affine cartesian codes
In this article, we give the answer to the following question: Given a field
, finite subsets of , and linearly
independent polynomials of total
degree at most . What is the maximal number of common zeros
can have in ? For , the
finite field with elements, answering this question is equivalent to
determining the generalized Hamming weights of the so-called affine Cartesian
codes. Seen in this light, our work is a generalization of the work of
Heijnen--Pellikaan for Reed--Muller codes to the significantly larger class of
affine Cartesian codes.Comment: 12 Page
Weierstrass semigroups on the Giulietti-Korchm\'aros curve
In this article we explicitly determine the structure of the Weierstrass
semigroups for any point of the Giulietti-Korchm\'aros curve
. We show that as the point varies, exactly three possibilities
arise: One for the -rational points (already known in the
literature), one for the -rational
points, and one for all remaining points. As a result, we prove a conjecture
concerning the structure of in case is an -rational point. As a corollary we also obtain that
the set of Weierstrass points of is exactly its set of
-rational points
Sub-quadratic Decoding of One-point Hermitian Codes
We present the first two sub-quadratic complexity decoding algorithms for
one-point Hermitian codes. The first is based on a fast realisation of the
Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer
algebra for polynomial-ring matrix minimisation. The second is a Power decoding
algorithm: an extension of classical key equation decoding which gives a
probabilistic decoding algorithm up to the Sudan radius. We show how the
resulting key equations can be solved by the same methods from computer
algebra, yielding similar asymptotic complexities.Comment: New version includes simulation results, improves some complexity
results, as well as a number of reviewer corrections. 20 page
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