arXiv subjects

Jade Nardi

Publications and source records attributed to Jade Nardi.

13 recordsLinked to original sources

arXiv2026

New perspectives for code locality in the rank metric

In coding theory, local recovery enables the efficient recovery of some part of (lost) coded data by accessing only a small number of other data entries. Locality was mostly but intensively studied for the recovery of individual symbols, that is, in the context of the Hamming metric. In this work, we propose a new definition of locality for general rank-metric codes. This definition differs from a previous work of Kadhe, El Rouayheb, Duursma and Sprintson [IEEE Trans. Inf. Theory 2019], by allowing to efficiently recover any element of the support, and without relying on any choice of bases of the underlying vector spaces. Our work firstly relies on a precise study of code puncturing and shortening for codes viewed as spaces of linear maps. We then provide examples and general constructions, showing the difference between our notion and that of Kadhe et al. We then derive a Singleton-like bound for rank locally recoverable codes, and we finally prove that a construction similar to classical Tamo-Barg codes is optimal with respect to this bound.

cs.IT↗

arXiv2026

Structure of weighted projective Reed-Muller codes

We provide a comprehensive overview of the fundamental structural properties of weighted projective Reed-Muller codes. We give a recursive construction for these codes, under some conditions for the weights, and we use it to derive bounds on the generalized Hamming weights and to obtain a recursive construction for their subfield subcodes and their dual codes. The dual codes are further studied in more generality, where the recursive constructions may not apply, obtaining a description as an evaluation code when the degree is low. We also provide insights into the Schur products of these codes when they are not degenerate.

cs.IT↗

arXiv2026

A computational approach to Drinfeld modules

This survey provides a practical and algorithmic perspective on Drinfeld modules over $\mathbb F_q[T]$. Starting with the construction of the Carlitz module, we present Drinfeld modules in any rank and some of their arithmetic properties. We emphasise the analogies with elliptic curves, and in the meantime, we also highlight key differences such as their rank structure and their associated Anderson motives. This document is designed for researchers in number theory, arithmetic geometry, algorithmic number theory, cryptography, or computer algebra, offering tools and insights to navigate the computational aspects of Drinfeld modules effectively. We include detailed SageMath implementations to illustrate explicit computations and facilitate experimentation. Applications to polynomial factorisation, isogeny computations, cryptographic constructions, and coding theory are also presented.

math.NT↗

arXiv2025

Maximum number of zeroes of polynomials on weighted projective spaces over a finite field

We compute the maximum number of rational points at which a homogeneous polynomial can vanish on a weighted projective space over a finite field, provided that the first weight is equal to one. This solves a conjecture by Aubry, Castryck, Ghorpade, Lachaud, O'Sullivan and Ram, which stated that a Serre-like bound holds with equality for weighted projective spaces when the first weight is one, and when considering polynomials whose degree is divisible by the least common multiple of the weights. We refine this conjecture by lifting the restriction on the degree and we prove it using footprint techniques, Delorme's reduction and Serre's classical bound.

math.AG↗

arXiv2025

A divide-and-conquer sumcheck protocol

We present a new sumcheck protocol called Fold-DCS (Fold-Divide-and-Conquer-Sumcheck) for multivariate polynomials based on a divide-and-conquer strategy. Its round complexity and soundness error are logarithmic in the number of variables, whereas they are linear in the classical sumcheck protocol. This drastic improvement in number of rounds and soundness comes at the expense of exchanging multivariate polynomials, which can be alleviated using polynomial commitment schemes. We first present Fold-DCS in the PIOP model, where the prover provides oracle access to a multivariate polynomial at each round. We then replace this oracle access in practice with a multivariate polynomial commitment scheme; we illustrate this with an adapted version of the recent commitment scheme Zeromorph [KT24], which allows us to replace most of the queries made by the verifier with a single batched evaluation check.

cs.IT↗

arXiv2024

Codes on Weighted Projective Planes

We comprehensively study weighted projective Reed-Muller (WPRM) codes on weighted projective planes $\mathbb{P}(1,a,b)$. We provide the universal Gr\"obner basis for the vanishing ideal of the set $Y$ of $\mathbb{F}_q$--rational points of $\mathbb{P}(1,a,b)$ to get the dimension of the code. We determine the regularity set of $Y$ using a novel combinatorial approach. We employ footprint techniques to compute the minimum distance.

math.AG↗

arXiv2023

Goppa-like AG codes from $C_{a,b}$ curves and their behaviour under squaring their dual

In this paper, we introduce a family of codes that can be used in a McEliece cryptosystem, called Goppa--like AG codes. These codes generalize classical Goppa codes and can be constructed from any curve of genus $\mathfrak{g} \geq 0$. Focusing on codes from $C_{a,b}$ curves, we study the behaviour of the dimension of the square of their dual to determine their resistance to distinguisher attacks similar to the one for alternant and Goppa codes developed by Mora and Tillich. We also propose numerical experiments to measure how sharp is our bound.

cs.IT↗

arXiv2022

Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code $C = C(\mathcal{X}, \mathcal{P}, D)$ over an algebraic curve $\mathcal{X}$ is a vector space associated to evaluations on $\mathcal{P}$ of functions in the Riemann-Roch space $L_\mathcal{X}(D)$. The problem of testing proximity to an error-correcting code $C$ consists in distinguishing between the case where an input word, given as an oracle, belongs to $C$ and the one where it is far from every codeword of $C$. AG codes are good candidates to construct short proof systems, but there exists no efficient proximity tests for them. We aim to fill this gap. We construct an Interactive Oracle Proof of Proximity (IOPP) for some families of AG codes by generalizing an IOPP for Reed-Solomon codes introduced by Ben-Sasson, Bentov, Horesh and Riabzev, known as the FRI protocol. We identify suitable requirements for designing efficient IOPP systems for AG codes. Our approach relies on a neat decomposition of the Riemann-Roch space of any invariant divisor under a group action on a curve into several explicit Riemann-Roch spaces on the quotient curve. We provide sufficient conditions on an AG code $C$ that allow to reduce a proximity testing problem for $C$ to a membership problem for a significantly smaller code $C'$. As concrete instantiations, we study AG codes on Kummer curves and curves in the Hermitian tower. The latter can be defined over polylogarithmic-size alphabet. We specialize the generic AG-IOPP construction to reach linear prover running time and logarithmic verification on Kummer curves, and quasilinear prover time with polylogarithmic verification on the Hermitian tower.

cs.IT↗

arXiv2022

Curves on Frobenius classical surfaces in $\mathbb{P}^3$ over finite fields

In this paper we give an upper bound on the number of rational points on an irreducible curve $C$ of degree $δ$ defined over a finite field $\mathbb{F}_q$ lying on a Frobenius classical surface $S$ embedded in $\mathbb{P}^3$. This leads us to investigate arithmetic properties of curves lying on surfaces. In a certain range of $δ$ and $q$, our result improves all other known bounds in the context of space curves.

math.AG↗

arXiv2021

Projective toric codes

Any integral convex polytope $P$ in $\mathbb{R}^N$ provides a $N$-dimensional toric variety $X_P$ and an ample divisor $D_P$ on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on $X_P$ , obtained by evaluating global section of $\mathcal{L}(D_P)$ on every rational point of $X_P$. This work presents an extension of toric codes analogous to the one of Reed-Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with non-zero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope $P$ and an algorithmic technique to get a lowerbound on the minimum distance is described.

math.AG↗

arXiv2019

Weighted Lifted Codes: Local Correctabilities and Application to Robust Private Information Retrieval

Low degree Reed-Muller codes are known to satisfy local decoding properties which find applications in private information retrieval (PIR) protocols, for instance. However, their practical instantiation encounters a first barrier due to their poor information rate in the low degree regime. This lead the community to design codes with similar local properties but larger dimension, namely the lifted Reed-Solomon codes. However, a second practical barrier appears when one requires that the PIR protocol resists collusions of servers. In this paper, we propose a solution to this problem by considering \emph{weighted} Reed-Muller codes. We prove that such codes allow us to build PIR protocols with optimal computation complexity and resisting to a small number of colluding servers. In order to improve the dimension of the codes, we then introduce an analogue of the lifting process for weigthed degrees. With a careful analysis of their degree sets, we notably show that the weighted lifting of Reed-Solomon codes produces families of codes with remarkable asymptotic parameters.

cs.IT↗

arXiv2019

Anticanonical codes from del Pezzo surfaces with Picard rank one

We construct algebraic geometric codes from del Pezzo surfaces and focus on the ones having Picard rank one and the codes associated to the anticanonical class. We give explicit constructions of del Pezzo surfaces of degree 4, 5 and 6, compute the parameters of the associated anticanonical codes and study their isomorphisms arising from the automorphisms of the surface. We obtain codes with excellent parameters and some of them turn out to beat the best known codes listed on the database codetable.

math.AG↗

arXiv2018

Algebraic Geometric codes on minimal Hirzebruch surfaces

We define a linear code $C_η(δ_T,δ_X)$ by evaluating polynomials of bidegree $(δ_T,δ_X)$ in the Cox ring on $\mathbb{F}_q$-rational points of the Hirzebruch surface of parameter $η$ on the finite field $\mathbb{F}_q$. We give explicit parameters of the code, notably using Gröbner bases. The minimum distance provides an upper bound of the number of $\mathbb{F}_q$-rational points of a non-filling curve on a Hirzebruch surface.

cs.IT↗