arXiv subjects
Publications and source records attributed to Maria Montanucci.
In this article, we describe a new maximal function field $G$ over the finite field $\mathbb{F}_{11^2}$ with $11^2$ elements and show that it cannot be obtained as a subfield of the Hermitian function field. This provides the first known $\mathbb{F}_{p^2}$-maximal function field that satisfies this property. Further, we compute the automorphism group of $G$. The new maximal function field was found using AI.
Algebraic Geometry (AG) codes (i.e. linear codes from algebraic function fields) in the Hamming metric were proposed by Goppa in 1980 and have been intensively studied ever since. Linearized Algebraic Geometry codes, the analogue of AG codes in the sum-rank metric, were instead introduced more recently [9], using quotients of the ring of Ore polynomials with coefficients in an algebraic function field. In this paper, we further investigate the results in [9], providing explicit, optimal and asymptotic constructions.
In this article we explicitly determine the Weierstrass semigroup at any place of some $\mathbb{F}_{q^2}$-maximal Fermat function fields $\mathcal{F}_m$, namely for $m=(q+1)/2$ and $m=(q+1)/3$. These famous function fields arise as Galois subfields of the Hermitian function field, and even though they have been intensively studied in the literature, the Weierstrass semigroup at every place is still not fully known. Surprisingly enough this problem is in fact quite involved and $\mathcal{F}_m$ has many different types of Weierstrass semigroups. Moreover, its set of Weierstrass places is much richer than its set of rational places.
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 authors found new examples of maximal function fields that are not isomorphic to subfields of the Hermitian function field. In this paper we construct analogous function fields $\tilde{\mathcal{Y}}_{n,s}$ and $\tilde{\mathcal{X}}_{n,s,a,b}$ as subfields of the second generalized GK function field (BM) and determine their automorphism groups. Using that the automorphism group is an invariant under isomorphism, we show that the function fields $\tilde{\mathcal{Y}}_{n,s}$ and ${\mathcal{Y}}_{n,s}$, as well as $\tilde{\mathcal{X}}_{n,s,a,b}$ and $\mathcal{X}_{n,s,a,b}$, are not isomorphic unless $m/s$ divides $q^2-q+1$ and $3$ divides $n$. In other words, the difference between the BM and GGS function fields can be found again at the level of the subfields that we consider.
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 $\mathbb{F}_{q^2}$-maximal function field $Z_3$ having the third largest genus, for $q \equiv 0 \pmod 3$. The cases $q \equiv 2 \pmod 3$ and $q \equiv 1 \pmod 3$ have been in fact analyzed in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], respectively. As in the other two cases, the function field $Z_3$ arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, $Z_3$ has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of $\mathbb{F}_{q^2}$-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, $\mathrm{Aut}(Z_3)$ is exactly the automorphism group inherited from the Hermitian function field, apart from the case $q=3$.
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, focusing on two regimes. First, we study the question of how well full-length RS codes perform against insertions and deletions. For 2-dimensional RS codes, we provide a complete characterization of codes that cannot correct even a single insertion or deletion. Furthermore, we prove that for sufficiently large field size~$q$, nearly all full-length $2$-dimensional RS codes can correct up to $(1 - \delta)q$ insertion and deletion errors for any $0 < \delta < 1$. Extending beyond the 2-dimensional case, we show that for any $k \ge 2$, there exists a full-length $k$-dimensional RS code capable of correcting $q / (10k)$ insertion and deletion errors, provided $q$ is large enough. Second, we focus on rate $1/2$ RS codes that can correct a single insertion or deletion error. We present a polynomial-time algorithm that constructs such codes over fields of size $q = \Theta(k^4)$. This result matches the existential bound given in \cite{con2023reed}.
Effective and reliable data retrieval is critical for the feasibility of DNA storage, and the development of random access efficiency plays a key role in its practicality and reliability. In this paper, we study the Random Access Problem, which asks to compute the expected number of samples one needs in order to recover an information strand. Unlike previous work, we took a geometric approach to the problem, aiming to understand which geometric structures lead to codes that perform well in terms of reducing the random access expectation (Balanced Quasi-Arcs). As a consequence, two main results are obtained. The first is a construction for $k=3$ that outperforms previous constructions aiming to reduce the random access expectation. The second, exploiting a result from~\cite{gruica2024reducing}, is the proof of a conjecture from~\cite{bar2023cover} for rate $1/2$ codes in any dimension.
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 $\mathcal{H}_q$ and $\mathcal{C}_d$ intersect in exactly $d(q+1)$ distinct $\mathbb{F}_{q^2}$-rational points? B\'ezout's theorem immediately implies that $\mathcal{H}_q$ and $\mathcal{C}_d$ intersect in at most $d(q+1)$ points, but equality is not guaranteed over $\mathbb{F}_{q^2}$. In this paper we prove that for many $d \le q^2-q+1$, the answer to this question is affirmative. The case $d=1$ is trivial: it is well known that any secant line of $\mathcal{H}_q$ defined over $\mathbb{F}_{q^2}$ intersects $\mathcal{H}_q$ in $q+1$ rational points. Moreover, all possible intersections of conics and $\mathcal{H}_q$ were classified by Donati et al. in 2009 and their results imply that the answer to the question above is affirmative for $d=2$ and $q \ge 4$, as well. However, an exhaustive computer search quickly reveals that for $(q,d) \in \{(2,2),(3,2),(2,3)\}$, the answer is instead negative. We show that for $q \le d \le q^2-q+1$, $d=\lfloor(q+1)/2\rfloor$ and $d=3$, $q \geq 3$ the answer is again affirmative. Various partial results for the case $d$ small compared to $q$ are also provided.
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 largest genus, for $q \equiv 1 \pmod 3$. This function field arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, $Y_3$ has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of $\mathbb{F}_{q^2}$-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, $\mathrm{Aut}(Y_3)$ is exactly the automorphism group inherited from the Hermitian function field, apart from small values of $q$.
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 function fields that arise as Galois subfields of the Hermitian function field. We compute the automorphism group, the Weierstrass semigroup at some special rational places and the isomorphism classes of such function fields. In this way, we show that often these function fields provide in fact examples of maximal function fields with the same genus, the same automorphism group, but that are not isomorphic.
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, but the genus $g$ of the used function field gives rise to negative term in the decoding radius, which we call the genus penalty. In this article, we present a GS-like list-decoding algorithm for arbitrary AG codes, which we call the \emph{inseparable GS list-decoder}. Apart from the multiplicity parameter $s$ and designed list size $\ell$, common for the GS list-decoder, we introduce an inseparability exponent $e$. Choosing this exponent to be positive gives rise to a list-decoder for which the genus penalty is reduced with a factor $1/p^e$ compared to the usual GS list-decoder. Here $p$ is the characteristic. Our list-decoder can be executed in $\tilde{\mathcal{O}}(s\ell^ωμ^{ω-1}p^e(n+g))$ field operations, where $n$ is the code length.
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 subcover of the Hermitian curve, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough $\mathcal{X}_3$ has many different types of Weierstrass semigroups and the set of its Weierstrass points is much richer than the set of $\mathbb{F}_{q^2}$-rational points, as instead happens for all the known maximal curves where the Weierstrass points are known. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, $\mathrm{Aut}(\mathcal{X}_3)$ is exactly the automorphism group inherited from the Hermitian curve, apart from small values of $q$.
In this paper we compute the automorphism group of the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ introduced in Tafazolian et al. in 2016 as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.
Let $\mathcal{X}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic $p$. Let $g$ and $γ$ be the genus and $p$-rank of $\mathcal{X}$, respectively. The influence of $g$ and $γ$ on the automorphism group $Aut(\mathcal{X})$ of $\mathcal{X}$ is well-known in the literature. If $g \geq 2$ then $Aut(\mathcal{X})$ is a finite group, and unless $\mathcal{X}$ is the so-called Hermitian curve, its order is upper bounded by a polynomial in $g$ of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth's bound of cube order in $g$ up to few exceptions, all having $p$-rank zero. In this paper a further refinement of Henn's result is proposed. First, we prove that if an algebraic curve of genus $g \geq 2$ has more than $336g^2$ automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame. Then we show that if $|Aut(\mathcal{X})| \geq 900g^2$, the quotient curve $\mathcal{X}/Aut(\mathcal{X})_P^{(1)}$ where $P$ is contained in the non-tame short orbit is rational, and the stabilizer of 2 points is either a $p$-group or a prime-to-$p$ group, then the $p$-rank of $\mathcal{X}$ is equal to zero.
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 respectively. The Ihara constant $A(q)$ has been intensively studied during the last decades, and it is defined as the limit superior of $|\mathcal{X}(\mathbb{F}_q)|/g(\mathcal X)$ as the genus of $\mathcal X$ goes to infinity. In 2012 Homma defined an analogue $D(q)$ of $A(q)$, where the nonsingularity of $\mathcal X$ is dropped and $g(\mathcal X)$ is replaced with the degree of $\mathcal X$. We will call $D(q)$ Homma's constant. In this paper, upper and lower bounds for the value of $D(q)$ are found.
Maximum scattered subspaces are not only objects of intrinsic interest in finite geometry but also powerful tools for the construction of MRD-codes, projective two-weight codes, and strongly regular graphs. In 2018 Csajbók, Marino, Polverino, and Zanella introduced a new family of maximum scattered subspaces in $\mathbb{F}_{q^6} \times \mathbb{F}_{q^6}$ arising from polynomials of type $f_b(x)=bx^q+x^{q^4}$ for certain choices of $b \in \mathbb{F}_{q^6}$. Throughout characterizations for $f_{b_2}(x)$ and $f_{b_1}(x)$ giving rise to equivalent maximum scattered subspaces, the authors conjectured that the portion of new and inequivalent maximum scattered subspaces obtained in this way is quite large. In this paper first we find necessary and sufficient conditions for $b$ to obtain a maximum scattered subspace. Such conditions were found independently with different techniques also by Polverino and Zullo 2019. Then we prove the conjecture on the number of new and inequivalent maximum scattered subspaces of this family.
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-Korchmáros cover of the Hermitian curve. In this paper a complete characterization of all Galois subcovers of the Skabelund curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ is given. Calculating the genera of the corresponding curves, we find new additions to the list of known genera of maximal curves over finite fields.
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 the previous bound $N_q(C) \leq (d-1)q+1$ obtained by Homma in 2012 and leads to a natural conjecture generalizing Sziklai's bound for the number of points of plane curves over finite fields.