arXiv subjects

Giacomo Micheli

Publications and source records attributed to Giacomo Micheli.

At least 19 recordsLinked to original sources

arXiv2026

A General Construction of Codes from Drinfeld Modules

We construct additive rank-metric and sum-rank-metric codes from Drinfeld modules by restricting bounded-degree morphisms to prime-to-characteristic torsion. For supersingular Drinfeld modules of rank $r$ in characteristic $\mathfrak{p}$ of degree $d$, the stabilization formula for morphism spaces yields rank-metric codes of $\mathbb{F}_q$-dimension $mrt-c$ and minimum distance $r-t+1$, where $c=r(r-1)(d-1)/2$. Simultaneous restriction to $\ell$ distinct degree-$m$ torsion modules gives additive sum-rank codes of the same dimension and minimum distance at least $\ell r-t+1$. Their normalized Singleton defects tend to zero, while in characteristic $(T)$ the module $ϕ_T=τ^r$ makes the defect vanish and produces an explicit MSRD family. We identify this family with a skew Chinese remainder theorem code supported on central skew polynomials and prove that its poly-skew weight is exactly $m$ times its sum-rank weight. This gives a specialized Singleton-type bound and a polynomial-time unique decoder up to the full sum-rank unique-decoding radius. We also derive a Welch-Berlekamp-type filter equation for the general supersingular sum-rank construction; it becomes an effective decoder whenever bases of the relevant morphism spaces and the restriction maps are computable.

math.NT↗

arXiv2026

Stabilization of isogeny spaces between supersingular Drinfeld modules

Let $\mathfrak{p}$ be a prime of degree $d$ in $A = \mathbb{F}_q[T]$ and let $ϕ, ψ$ be supersingular Drinfeld modules of rank $r \geq 2$ in $A$-characteristic $\mathfrak{p}$. We study the $\mathbb{F}_q$-dimension of the space $M_s(ϕ, ψ) = \{u \in \mathrm{Hom}(ϕ, ψ) : \mathrm{deg}_τu \leq s\}$ as a function of $s$. By analyzing $\mathrm{Hom}(ϕ, ψ)$ as a normed $A$-lattice in the local division algebra at $\infty$ via its successive minima, we obtain an exact closed-form expression for $\dim_{\mathbb{F}_q} M_s(ϕ, ψ)$ valid for every $s \geq 0$, together with structural constraints on the successive-minima multiset which imply the stabilization formula $\dim_{\mathbb{F}_q} M_s(ϕ, ψ) = r(s+1) - \frac{r(r-1)(d-1)}{2}$ for all $s \geq r^2(r-1)(d-1)/2$. We conjecture that the optimal threshold is $s \geq (r-1)(d-1) - 1$, and prove this sharp form for $r = 2$ by independent automorphic methods, using the decomposition of a Brandt-type theta series on the Bruhat-Tits tree of $\mathrm{PGL}_2(F_\infty)$ into Eisenstein and cuspidal parts together with the polynomiality of the cuspidal $L$-function. We also recast our results in Mornev's geometric framework, in which the conjecture becomes a cohomology-vanishing statement for a family of vector bundles on $\mathbb{P}^1$, and illustrate the theory with explicit examples in which all successive-minima multisets permitted by our constraints are realized.

math.NT↗

arXiv2026

A New Class of Linear Codes

Let $n$ be a prime power, $r$ be a prime with $r\mid n-1$, and $\varepsilon\in (0,1/2)$. Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over $\mathbb F_r$ having length $n$, relative distance $(r-1)/r+O(n^{-\varepsilon})$ and rate $n^{-1/2-\varepsilon}$. When $r=2$, our binary codes have exponential size when compared to all previously known families of linear and non-linear codes with relative distance asymptotic to $1/2$, such as Delsarte--Goethals codes. Moreover, concatenating with a Reed-Solomon code we get a family of codes of length $n$ and rate $n^{-1/(2n+2)-2\varepsilon/(n+1)}+O(n^{-1/(n+1)})$ and relative distance $1/2+O(n^{-\varepsilon})$. This shows that, for a fixed length, the rate of the concatenation suggested by Kschischang and Tasbihi (2024) of a Reed-Solomon and a Reed-Muller code can be made an order of magnitude smaller than a concatenation of a Reed-Solomon with a large dimensional Shadow code, while still keeping the regime of relative distance $1/2$. Finally, we show that the square of a Shadow code behaves like a random code and the Shadow code itself has a decoding algorithm, which suggest that such class of codes has the potential to be interesting for cryptographic applications.

cs.IT↗

arXiv2026

Rank metric codes from Drinfeld modules

We establish a connection between Drinfeld modules and rank-metric codes, focusing on the case of semifield codes. Our method constructs rank-metric codes from linear subspaces of endomorphisms of a Drinfeld module acting on torsion submodules. We show that Sheekey's construction [She20] fits naturally into this framework, yielding a short conceptual proof of one of his main results. We then give a new construction of infinite families of semifield codes arising from Drinfeld modules defined over finite fields.

math.NT↗

arXiv2026

Optimal Rank-Metric Codes with Rank-Locality from Drinfeld Modules

We introduce a new technique to construct rank-metric codes using the arithmetic theory of Drinfeld modules over global fields, and Dirichlet Theorem on polynomial arithmetic progressions. Using our methods, we obtain a new infinite family of optimal rank-metric codes with rank-locality, i.e. every code in our family achieves the information theoretical bound for rank-metric codes with rank-locality.

cs.IT↗

arXiv2025

On the Riemann Hypothesis for Drinfeld Modules

In this paper we provide a short proof of the Riemann Hypothesis for Drinfeld modules which uses only basic notions from the theory of global function fields and of Drinfeld modules.

math.NT↗

arXiv2025

On the Characteristic Polynomial of Linearized Polynomials

Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sum^r_{i=0}a_iZ^{q^i}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large extension field $\mathbb F_{q^n}\supseteq k$. Our algorithm has computational complexity of $O(n(\log(n))^4)$ in terms of $\mathbb F_q$ operations with the implied constant depending only on $k$ and $r$. Up to logarithmic factors, and for linear maps represented by low degree polynomials, this provides a square root improvement over generic algorithms.

math.NT↗

arXiv2024

Codes from $A_m$-invariant polynomials

Let $q$ be a prime power. This paper provides a new class of linear codes that arises from the action of the alternating group on $\mathbb F_q[x_1,\dots,x_m]$ combined with the ideas in (M. Datta and T. Johnsen, 2022). Compared with Generalized Reed-Muller codes with similar parameters, our codes have the same asymptotic relative distance but a better rate. Our results follow from combinations of Galois theoretical methods with Weil-type bounds for the number of points of hypersurfaces over finite fields.

cs.IT↗

arXiv2024

Square patterns in dynamical orbits

Let $q$ be an odd prime power. Let $f\in \mathbb{F}_q[x]$ be a polynomial having degree at least $2$, $a\in \mathbb{F}_q$, and denote by $f^n$ the $n$-th iteration of $f$. Let $χ$ be the quadratic character of $\mathbb{F}_q$, and $\mathcal{O}_f(a)$ the forward orbit of $a$ under iteration by $f$. Suppose that the sequence $(χ(f^n(a)))_{n\geq 1}$ is periodic, and $m$ is its period. Assuming a mild and generic condition on $f$, we show that, up to a constant, $m$ can be bounded from below by $|\mathcal{O}_f(a)|/q^\frac{2\log_{2}(d)+1}{2\log_2(d)+2}$. More informally, we prove that the period of the appearance of squares in an orbit of an element provides an upper bound for the size of the orbit itself. Using a similar method, we can also prove that, up to a constant, we cannot have more than $q^\frac{2\log_2(d)+1}{2\log_2(d)+2}$ consecutive squares or non-squares in the forward orbit of $a$. In addition, we provide a classification of all polynomials for which our generic condition does not hold.

math.NT↗

arXiv2023

Number Theoretical Locally Recoverable Codes

In this paper we give constructions for infinite sequences of finite non-linear locally recoverable codes $\mathcal C\subseteq \prod\limits^N_{i=1}\mathbb F_{q_i}$ over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity.

cs.IT↗

arXiv2023

On complete $m$-arcs

Let $m$ be a positive integer and $q$ be a prime power. For large finite base fields $\mathbb F_q$, we show that any curve can be used to produce a complete $m$-arc as long as some generic explicit geometric conditions on the curve are verified. To show the effectiveness of our theory, we derive complete $m$-arcs from hyperelliptic curves and from Artin-Schreier curves.

math.CO↗

arXiv2022

Differential biases, $c$-differential uniformity, and their relation to differential attacks

Differential cryptanalysis famously uses statistical biases in the propagation of differences in a block cipher to attack the cipher. In this paper, we investigate the existence of more general statistical biases in the differences. To this end, we discuss the $c$-differential uniformity of S-boxes, which is a concept that was recently introduced in Ellingsen et. al. to measure certain statistical biases that could potentially be used in attacks similar to differential attacks. Firstly, we prove that a large class of potential candidates for S-boxes necessarily has large $c$-differential uniformity for all but at most $B$ choices of $c$, where $B$ is a constant independent of the size of the finite field $q$. This result implies that for a large class of functions, certain statistical differential biases are inevitable. In a second part, we discuss the practical possibility of designing a differential attack based on weaknesses of S-boxes related to their $c$-differential uniformity.

cs.CR↗

arXiv2022

Optimal locally recoverable codes with hierarchy from nested $F$-adic expansions

In this paper we construct new optimal hierarchical locally recoverable codes. Our construction is based on a combination of the ideas of \cite{ballentine2019codes,sasidharan2015codes} with an algebraic number theoretical approach that allows to give a finer tuning of the minimum distance of the intermediate code (allowing larger dimension of the final code), and to remove restrictions on the arithmetic properties of $q$ compared with the size of the locality sets in the hierarchy. In turn, we manage to obtain codes with a wide set of parameters both for the size $q$ of the base field, and for the hierarchy size, while keeping the optimality of the codes we construct.

cs.IT↗

arXiv2022

Local to global principle over number fields for higher moments

The local to global principle for densities is a very convenient tool proposed by Poonen and Stoll to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a local to global principle that allows the computation of all higher moments corresponding to the density, over a general number field $K$. This work advances the understanding of local to global principles for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the local to global principle for densities over number fields; on the other hand, it extends the recent result on a local to global principle for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.

math.NT↗

arXiv2021

Optimal Selection for Good Polynomials of Degree up to Five

Good polynomials are the fundamental objects in the Tamo-Barg constructions of Locally Recoverable Codes (LRC). In this paper we classify all good polynomials up to degree $5$, providing explicit bounds on the maximal number $\ell$ of sets of size $r+1$ where a polynomial of degree $r+1$ is constant, up to $r=4$. This directly provides an explicit estimate (up to an error term of $O(\sqrt{q})$, with explict constant) for the maximal length and dimension of a Tamo-Barg LRC. Moreover, we explain how to construct good polynomials achieving these bounds. Finally, we provide computational examples to show how close our estimates are to the actual values of $\ell$, and we explain how to obtain the best possible good polynomials in degree $5$.

cs.IT↗

arXiv2020

$r$-fat linearized polynomials over finite fields

In this paper we prove that the property of being scattered for a $\mathbb{F}_q$-linearized polynomial of small $q$-degree over a finite field $\mathbb{F}_{q^n}$ is unstable, in the sense that, whenever the corresponding linear set has at least one point of weight larger than one, the polynomial is far from being scattered. To this aim, we define and investigate $r$-fat polynomials, a natural generalization of scattered polynomials. An $r$-fat $\mathbb{F}_q$-linearized polynomial defines a linear set of rank $n$ in the projective line of order $q^n$ with $r$ points of weight larger than one. When $r$ equals $1$, the corresponding linear sets are called clubs, and they are related with a number of remarkable mathematical objects like KM-arcs, group divisible designs and rank metric codes. Using techniques on algebraic curves and global function fields, we obtain numerical bounds for $r$ and the non-existence of exceptional $r$-fat polynomials with $r>0$. In the case $n\leq 4$, we completely determine the spectrum of values of $r$ for which an $r$-fat polynomial exists. In the case $n=5$, we provide a new family of $1$-fat polynomials. Furthermore, we determine the values of $r$ for which the so-called LP-polynomials are $r$-fat.

math.CO↗

arXiv2020

Local to global principle for expected values

This paper constructs a new local to global principle for expected values over free $\mathbb{Z}$-modules of finite rank. In our strategy we use the same philosophy as Ekedhal's Sieve for densities, later extended and improved by Poonen and Stoll in their local to global principle for densities. We show that under some additional hypothesis on the system of $p$-adic subsets used in the principle, one can use $p$-adic measures also when one has to compute expected values (and not only densities). Moreover, we show that our additional hypotheses are sharp, in the sense that explicit counterexamples exist when any of them is missing. In particular, a system of $p$-adic subsets that works in the Poonen and Stoll principle is not guaranteed to work when one is interested in expected values instead of densities. Finally, we provide both new applications of the method, and immediate proofs for known results.

math.NT↗

arXiv2020

Algebraic constructions of complete $m$-arcs

Let $m$ be a positive integer, $q$ be a prime power, and $\mathrm{PG}(2,q)$ be the projective plane over the finite field $\mathbb F_q$. Finding complete $m$-arcs in $\mathrm{PG}(2,q)$ of size less than $q$ is a classical problem in finite geometry. In this paper we give a complete answer to this problem when $q$ is relatively large compared with $m$, explicitly constructing the smallest $m$-arcs in the literature so far for any $m\geq 8$. For any fixed $m$, our arcs $\mathcal A_{q,m}$ satisfy $|\mathcal A_{q,m}|-q\rightarrow -\infty$ as $q$ grows. To produce such $m$-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the $m$-completeness of the arc.

math.CO↗