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Peter Beelen

Publications and source records attributed to Peter Beelen.

At least 55 records · Page 3Linked to original sources
arXiv2017

Weierstrass semigroups on the Giulietti-Korchmáros curve

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchmáros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the $\mathbb{F}_{q^2}$-rational points (already known in the literature), one for the $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational points, and one for all remaining points. As a result, we prove a conjecture concerning the structure of $H(P)$ in case $P$ is an $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational point. As a corollary we also obtain that the set of Weierstrass points of $\mathcal{X}$ is exactly its set of $\mathbb{F}_{q^6}$-rational points.

math.AG↗
arXiv2017

Generalized Hamming weights of affine cartesian codes

In this article, we give the answer to the following question: Given a field $\mathbb{F}$, finite subsets $A_1,\dots,A_m$ of $\mathbb{F}$, and $r$ linearly independent polynomials $f_1,\dots,f_r \in \mathbb{F}[x_1,\dots,x_m]$ of total degree at most $d$. What is the maximal number of common zeros $f_1,\dots,f_r$ can have in $A_1 \times \cdots \times A_m$? For $\mathbb{F}=\mathbb{F}_q$, the finite field with $q$ 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.

math.AG↗
arXiv2017

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 $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables with coefficients in a finite field with $q$ elements can have in the corresponding $m$-dimensional projective space. Recently, it has been shown by Datta and Ghorpade that this conjecture is valid if $r$ is at most $m+1$ and can be invalid otherwise. Moreover a new conjecture was proposed for many values of $r$ beyond $m+1$. In this paper, we prove that this new conjecture holds true for several values of $r$. In particular, this settles the new conjecture completely when $d=3$. 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 $d=q-1$ and of $d=q$. 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.

math.AG↗
arXiv2017

Twisted Reed-Solomon Codes

We present a new general construction of MDS codes over a finite field $\mathbb{F}_q$. We describe two explicit subclasses which contain new MDS codes of length at least $q/2$ for all values of $q \ge 11$. Moreover, we show that most of the new codes are not equivalent to a Reed-Solomon code.

cs.IT↗
arXiv2016

Counting generalized Reed-Solomon codes

In this article we count the number of generalized Reed-Solomon (GRS) codes of dimension k and length n, including the codes coming from a non-degenerate conic plus nucleus. We compare our results with known formulae for the number of 3-dimensional MDS codes of length n=6,7,8,9.

cs.IT↗
arXiv2016

Good families of Drinfeld modular curves

In this paper we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach on how to obtain explicit defining equations for some of these towers and in particular give a new explicit example of an optimal tower over a quadratic finite field.

math.NT↗
arXiv2015

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.

cs.IT↗
arXiv2015

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.

math.CO↗
arXiv2013

Galois Towers over Non-prime Finite Fields

In this paper we construct Galois towers with good asymptotic properties over any non-prime finite field $\mathbb F_{\ell}$; i.e., we construct sequences of function fields $\mathcal{N}=(N_1 \subset N_2 \subset \cdots)$ over $\mathbb F_{\ell}$ of increasing genus, such that all the extensions $N_i/N_1$ are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with these properties are important for applications in various fields including coding theory and cryptography.

math.AG↗
arXiv2013

Good Towers of Function Fields

In this paper, we will give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory and the Drinfeld modular theory). In the classical modular setup, optimal towers can be obtained, while in the Drinfeld modular setup, good towers over any non-prime field may be found. We illustrate the theory with several examples, thus explaining some known towers as well as giving new examples of good explicitly defined towers of function fields.

math.NT↗
arXiv2013

Towers of Function Fields over Non-prime Finite Fields

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity $A(\ell)$, for $\ell = p^n$ with $p$ prime and $n>3$ odd. We relate the explicit equations to Drinfeld modular varieties.

math.AG↗
arXiv2013

On Rational-Interpolation Based List-Decoding and List-Decoding Binary Goppa Codes

We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm (EA) as the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm (GSA), immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes.

cs.IT↗
arXiv2012

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. Tutaş: A generalization of the Weierstrass semigroup, J. Pure Appl. Algebra, 207(2), 2006] for an $n$-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 $\mathbb{F}_q$-rational places in algebraic function fields using Weierstrass semigroups. Pure Appl. Algebra, 213(6), 2009].

math.AG↗
arXiv2011

Explicit equations for Drinfeld modular towers

Elaborating on ideas of Elkies, we show how recursive equations for towers of Drinfeld modular curves $(X_0(P^n))_{n\ge 0}$ for $P\in \mathbb F_q[T]$ can be read of directly from the modular polynomial $Φ_P(X,Y)$ and how this naturally leads to recursions of depth two. Although the modular polynomial $Φ_T(X,Y)$ is not known in general, using generators and relations given by Schweizer, we find unreduced recursive equations over $\mathbb F_q(T)$ for the tower $(X_0(T^n))_{n\ge 2}$ and of a small variation of it (its partial Galois closure). Reducing at various primes, one obtains towers over finite fields, which are optimal, i.e., reach the Drinfeld--Vladut bound, over a quadratic extension of the finite field. We give a proof of the optimality of these towers, which is elementary and does not rely on their modular interpretation except at one point. We employ the modular interpretation to determine the splitting field of certain polynomials, which are analogues of the Deuring polynomial. For these towers, the particular case of reduction at the prime $T-1$ corresponds to towers introduced by Elkies and Garcia--Stichtenoth.

math.NT↗
arXiv2011

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.

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