Peter Beelen
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View article: Non-isomorphic subfields of the BM and GGS maximal function fields
Non-isomorphic subfields of the BM and GGS maximal function fields Open
In 2016 Tafazolian et al. introduced new families of $\mathbb{F}_{q^{2n}}$-maximal function fields $\mathcal{Y}_{n,s}$ and $\mathcal{X}_{n,s,a,b}$ arising as subfields of the first generalized GK function field (GGS). In this way the autho…
View article: Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$
Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$ Open
In this article we complete the work started in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known $\mathbb{F}_{q^2}$-maxi…
View article: Reed-Solomon Codes Against Insertions and Deletions: Full-Length and Rate-$1/2$ Codes
Reed-Solomon Codes Against Insertions and Deletions: Full-Length and Rate-$1/2$ Codes Open
The performance of Reed--Solomon codes (RS codes, for short) in the presence of insertion and deletion errors has attracted growing attention in recent literature. In this work, we further study this intriguing mathematical problem, focusi…
View article: A family of non-isomorphic maximal function fields
A family of non-isomorphic maximal function fields Open
The problem of understanding whether two given function fields are isomorphic is well-known to be difficult, particularly when the aim is to prove that an isomorphism does not exist. In this paper we investigate a family of maximal functio…
View article: Linear codes associated to symmetric determinantal varieties; General case
Linear codes associated to symmetric determinantal varieties; General case Open
The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance …
View article: Intersection of irreducible curves and the Hermitian curve
Intersection of irreducible curves and the Hermitian curve Open
Let $\mathcal{H}_q$ denote the Hermitian curve in $\mathbb{P}^2$ over $\mathbb{F}_{q^2}$ and $\mathcal{C}_d$ be an irreducible plane projective curve in $\mathbb{P}^2$ also defined over $\mathbb{F}_{q^2}$ of degree $d$. Can $\mathcal{H}_q$…
View article: Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$
Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$ Open
In this article we continue the work started in arXiv:2303.00376v1, explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known $\mathbb{F}_{q^2}$-maximal function field $Y_3$ having the third l…
View article: Some families of non-isomorphic maximal function fields
Some families of non-isomorphic maximal function fields Open
The problem of understanding whether two given function fields are isomorphic is well-known to be difficult, particularly when the aim is to prove that an isomorphism does not exist. In this paper we investigate a family of maximal functio…
View article: Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus
Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus Open
In this article we explicitly determine the Weierstrass semigroup at any point and the full automorphism group of a known Fq2-maximal curve X3 having the third largest genus. This curve arises as a Galois subcover of the Hermitian curve, a…
View article: List-decoding of AG codes without genus penalty
List-decoding of AG codes without genus penalty Open
In this paper we consider algebraic geometry (AG) codes: a class of codes constructed from algebraic codes (equivalently, using function fields) by Goppa. These codes can be list-decoded using the famous Guruswami-Sudan (GS) list-decoder, …
View article: Faster List Decoding of AG Codes
Faster List Decoding of AG Codes Open
In this article, we present a fast algorithm performing an instance of the Guruswami-Sudan list decoder for algebraic geometry codes. We show that any such code can be decoded in $\tilde{O}(s^2\ell^{ω-1}μ^{ω-1}(n+g) + \ell^ωμ^ω)$ operation…
View article: Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus
Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus Open
In this article we explicitly determine the Weierstrass semigroup at any point and the full automorphism group of a known $\mathbb{F}_{q^2}$-maximal curve $\mathcal{X}_3$ having the third largest genus. This curve arises as a Galois subcov…
View article: Fast Decoding of AG Codes
Fast Decoding of AG Codes Open
We present an efficient list decoding algorithm in the style of Guruswami-Sudan for algebraic geometry codes. Our decoder can decode any such code using $\tilde{\mathcal O}(s\ell^ωμ^{ω-1}(n+g))$ operations in the underlying finite field, w…
View article: A survey on recursive towers and Ihara's constant
A survey on recursive towers and Ihara's constant Open
Since Serre gave his famous Harvard lectures in 1985 on various aspects of the theory of algebraic curves defined over a finite field, there have been many developments. In this survey article, an overview will be given on the developments…
View article: Twisted Reed–Solomon Codes
Twisted Reed–Solomon Codes Open
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 th…
View article: On the constant $D(q)$ defined by Homma
On the constant $D(q)$ defined by Homma Open
Let $\mathcal{X}$ be a projective, irreducible, nonsingular algebraic curve over the finite field $\mathbb{F}_q$ with $q$ elements and let $|\mathcal{X}(\mathbb{F}_q)|$ and $g(\mathcal X)$ be its number of rational points and genus respect…
View article: A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields
A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields Open
We give a complete conjectural formula for the number er(d, m) of maximum possible Fq-rational points on a projective algebraic variety defined by r linearly independent homogeneous polynomial equations of degree d in m + 1 variables with …
View article: Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case
Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case Open
We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have co…
View article: Classification of all Galois subcovers of the Skabelund maximal curves
Classification of all Galois subcovers of the Skabelund maximal curves Open
In 2017 Skabelund constructed two new examples of maximal curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the Giulietti-Korchm…
View article: Fast Encoding of AG Codes Over C<sub>ab</sub>Curves
Fast Encoding of AG Codes Over C<sub>ab</sub>Curves Open
We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called C{ab} curves, as well as algorithms for inverting the encoding map, which we call 'unencoding'. Some C{ab} curves have many …
View article: A bound for the number of points of space curves over finite fields
A bound for the number of points of space curves over finite fields Open
For a non-degenerate irreducible curve $C$ of degree $d$ in $\mathbb{P}^3$ over $\mathbb{F}_q$, we prove that the number $N_q(C)$ of $\mathbb{F}_q$-rational points of $C$ satisfies the inequality $N_q(C) \leq (d-2)q+1$. Our result improves…
View article: Maximum Number of Points on Intersection of a Cubic Surface and a Non-Degenerate Hermitian Surface
Maximum Number of Points on Intersection of a Cubic Surface and a Non-Degenerate Hermitian Surface Open
In 1991 Sørensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in the case whe…