1,675 research outputs found
On triply even binary codes
A triply even code is a binary linear code in which the weight of every
codeword is divisible by 8. We show how two doubly even codes of lengths m_1
and m_2 can be combined to make a triply even code of length m_1+m_2, and then
prove that every maximal triply even code of length 48 can be obtained by
combining two doubly even codes of length 24 in a certain way. Using this
result, we show that there are exactly 10 maximal triply even codes of length
48 up to equivalence.Comment: 21 pages + appendix of 10 pages. Minor revisio
Two-point coordinate rings for GK-curves
Giulietti and Korchm\'aros presented new curves with the maximal number of
points over a field of size q^6. Garcia, G\"uneri, and Stichtenoth extended the
construction to curves that are maximal over fields of size q^2n, for odd n >=
3. The generalized GK-curves have affine equations x^q+x = y^{q+1} and
y^{q^2}-y^q = z^r, for r=(q^n+1)/(q+1). We give a new proof for the maximality
of the generalized GK-curves and we outline methods to efficiently obtain their
two-point coordinate ring.Comment: 16 page
Partial Spreads in Random Network Coding
Following the approach by R. K\"otter and F. R. Kschischang, we study network
codes as families of k-dimensional linear subspaces of a vector space F_q^n, q
being a prime power and F_q the finite field with q elements. In particular,
following an idea in finite projective geometry, we introduce a class of
network codes which we call "partial spread codes". Partial spread codes
naturally generalize spread codes. In this paper we provide an easy description
of such codes in terms of matrices, discuss their maximality, and provide an
efficient decoding algorithm
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
Structure and Interpretation of Dual-Feasible Functions
We study two techniques to obtain new families of classical and general
Dual-Feasible Functions: A conversion from minimal Gomory--Johnson functions;
and computer-based search using polyhedral computation and an automatic
maximality and extremality test.Comment: 6 pages extended abstract to appear in Proc. LAGOS 2017, with 21
pages of appendi
Prefactor Reduction of the Guruswami-Sudan Interpolation Step
The concept of prefactors is considered in order to decrease the complexity
of the Guruswami-Sudan interpolation step for generalized Reed-Solomon codes.
It is shown that the well-known re-encoding projection due to Koetter et al.
leads to one type of such prefactors. The new type of Sierpinski prefactors is
introduced. The latter are based on the fact that many binomial coefficients in
the Hasse derivative associated with the Guruswami-Sudan interpolation step are
zero modulo the base field characteristic. It is shown that both types of
prefactors can be combined and how arbitrary prefactors can be used to derive a
reduced Guruswami-Sudan interpolation step.Comment: 13 pages, 3 figure
On maximal curves
We study arithmetical and geometrical properties of maximal curves, that is,
curves defined over the finite field F_{q^2} whose number of F_{q^2}-rational
points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a
rational point, we prove that maximal curves are F_{q^2}-isomorphic to y^q + y
= x^m, for some . As a consequence we show that a maximal curve of
genus g=(q-1)^2/4 is F_{q^2}-isomorphic to the curve y^q + y = x^{(q+1)/2}.Comment: LaTex2e, 17 pages; this article is an improved version of the paper
alg-geom/9603013 (by Fuhrmann and Torres
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