1,314 research outputs found
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
Affine Grassmann Codes
We consider a new class of linear codes, called affine Grassmann codes. These
can be viewed as a variant of generalized Reed-Muller codes and are closely
related to Grassmann codes. We determine the length, dimension, and the minimum
distance of any affine Grassmann code. Moreover, we show that affine Grassmann
codes have a large automorphism group and determine the number of minimum
weight codewords.Comment: Slightly Revised Version; 18 page
Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields
About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the
maximum number of common zeros that linearly independent homogeneous
polynomials of degree in variables with coefficients in a finite
field with elements can have in the corresponding -dimensional
projective space. Recently, it has been shown by Datta and Ghorpade that this
conjecture is valid if is at most and can be invalid otherwise.
Moreover a new conjecture was proposed for many values of beyond . In
this paper, we prove that this new conjecture holds true for several values of
. In particular, this settles the new conjecture completely when . Our
result also includes the positive result of Datta and Ghorpade as a special
case. Further, we determine the maximum number of zeros in certain cases not
covered by the earlier conjectures and results, namely, the case of and
of . All these results are directly applicable to the determination of the
maximum number of points on sections of Veronese varieties by linear
subvarieties of a fixed dimension, and also the determination of generalized
Hamming weights of projective Reed-Muller codes.Comment: 15 page
Linear Codes associated to Determinantal Varieties
We consider a class of linear codes associated to projective algebraic
varieties defined by the vanishing of minors of a fixed size of a generic
matrix. It is seen that the resulting code has only a small number of distinct
weights. The case of varieties defined by the vanishing of 2 x 2 minors is
considered in some detail. Here we obtain the complete weight distribution.
Moreover, several generalized Hamming weights are determined explicitly and it
is shown that the first few of them coincide with the distinct nonzero weights.
One of the tools used is to determine the maximum possible number of matrices
of rank 1 in a linear space of matrices of a given dimension over a finite
field. In particular, we determine the structure and the maximum possible
dimension of linear spaces of matrices in which every nonzero matrix has rank
1.Comment: 12 pages; to appear in Discrete Mat
Bounding the number of points on a curve using a generalization of Weierstrass semigroups
In this article we use techniques from coding theory to derive upper bounds
for the number of rational places of the function field of an algebraic curve
defined over a finite field. The used techniques yield upper bounds if the
(generalized) Weierstrass semigroup [P. Beelen, N. Tuta\c{s}: A generalization
of the Weierstrass semigroup, J. Pure Appl. Algebra, 207(2), 2006] for an
-tuple of places is known, even if the exact defining equation of the curve
is not known. As shown in examples, this sometimes enables one to get an upper
bound for the number of rational places for families of function fields. Our
results extend results in [O. Geil, R. Matsumoto: Bounding the number of
-rational places in algebraic function fields using Weierstrass
semigroups. Pure Appl. Algebra, 213(6), 2009]
Automorphism groups of Grassmann codes
We use a theorem of Chow (1949) on line-preserving bijections of
Grassmannians to determine the automorphism group of Grassmann codes. Further,
we analyze the automorphisms of the big cell of a Grassmannian and then use it
to settle an open question of Beelen et al. (2010) concerning the permutation
automorphism groups of affine Grassmann codes. Finally, we prove an analogue of
Chow's theorem for the case of Schubert divisors in Grassmannians and then use
it to determine the automorphism group of linear codes associated to such
Schubert divisors. In the course of this work, we also give an alternative
short proof of MacWilliams theorem concerning the equivalence of linear codes
and a characterization of maximal linear subspaces of Schubert divisors in
Grassmannians.Comment: revised versio
Transaction Costs for Design-Build-Finance-Maintain Contracts
This paper gives insight in how transaction costs arise and how in theory transaction costs can be reduced. A comparison between theory and practice has been made. A study of a case in the Netherlands, the Second Coentunnel showed how transaction costs in practice appear, in which stage of the purchasing process these cost arise and also how transaction costs can be reduced. Cost specifications, handed by the public and private parties, make clear that in every phase of the process the client makes expenses. The client spends the most money during the initiative phase. The private parties start making costs in the first phase of the tender (prequalification). For contractors the most expensive phase is the dialogue phase. Taking all the costs in overview, noticeable is that all of the costs are related to the duration of the different phases of the process and required capacity of personnel. Success factors from theory and practice have been identified in the process in which transaction costs arise. Theory and practice have been compared and resulted in a list of twelve success factors. By implementing these success factors in future projects the expectation is that transaction costs will not be unnecessary high
Duals of Affine Grassmann Codes and their Relatives
Affine Grassmann codes are a variant of generalized Reed-Muller codes and are
closely related to Grassmann codes. These codes were introduced in a recent
work [2]. Here we consider, more generally, affine Grassmann codes of a given
level. We explicitly determine the dual of an affine Grassmann code of any
level and compute its minimum distance. Further, we ameliorate the results of
[2] concerning the automorphism group of affine Grassmann codes. Finally, we
prove that affine Grassmann codes and their duals have the property that they
are linear codes generated by their minimum-weight codewords. This provides a
clean analogue of a corresponding result for generalized Reed-Muller codes.Comment: 20 page
Vanishing ideals of projective spaces over finite fields and a projective footprint bound
We consider the vanishing ideal of a projective space over a finite field. An
explicit set of generators for this ideal has been given by Mercier and
Rolland. We show that these generators form a universal Gr\"obner basis of the
ideal. Further we give a projective analogue of the footprint bound, and a
version of it that is suitable for estimating the number of points of a
projective algebraic variety over a finite field. An application to Serre's
inequality for the number of rational points of projective hypersurfaces over
finite fields is includedComment: 16 pages, slightly revised version, to appear in Acta Math. Sin.
(Engl. Ser.
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