arXiv subjects

Adler Marques

Publications and source records attributed to Adler Marques.

5 recordsLinked to original sources

arXiv2026

Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation

We study algebraic geometry codes on hyperelliptic curves of genus $g \geq 2$ with complementarity properties. Our first contribution is a characterization of non-special divisors of degree $g$ and $g-1$ via the polynomial degrees of their reduced Mumford representation, reducing a classical hard geometric problem to a single-degree test on univariate polynomials. Using this, we construct Linear Complementary Pairs (LCP) of codes via polynomial arithmetic on the Jacobian and provide a criterion in terms of Mumford degrees for the resulting codes to be Maximum Distance Separable (MDS). Under a $2$-torsion condition in the Jacobian, equivalently a divisibility condition on the Mumford polynomials, we obtain explicit multipliers that turn these pairs into Linear Complementary Dual (LCD) codes. Finally, we apply this framework to the maximal hyperelliptic curve $\mathcal{X} \colon y^2 = x^q + x$ over $\mathbb{F}_{q^2}$ and give explicit examples of MDS LCD codes with parameters $[2q,q,q+1]_{q^2}$ for $q = 4, 5, 7$, verified computationally; we conjecture, with heuristic support, that such codes exist for all $q \geq 4$.

math.AG↗

arXiv2026

On planar functions over $\mathbb{F}_{q^3}$

Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.

math.NT↗

arXiv2026

Construction of Non-special Divisors on Kummer Covers With Arbritary Ramification For LCP Codes

Linear Complementary Pairs (LCP) of algebraic geometry (AG) codes offer strong resistance against side-channel and fault-injection attacks, but their construction depends critically on the explicit identification of non-special divisors of degree $g$ and $g-1$. Existing constructions are restricted to Kummer extensions where divisors are supported exclusively on totally ramified places, significantly limiting the range of applicable function fields and codes. We remove this restriction by developing a framework for general Kummer extensions $y^m = \prod_{i=1}^r (x-\alpha_i)^{\lambda_i}$ over finite fields with arbitrary ramification. Using Galois group actions and invariant divisor techniques, we establish necessary and sufficient conditions for non-speciality with no constraint on the support, yielding explicit constructions where previous methods fail. Our approach replaces the computationally intensive Weierstrass semigroup machinery with a more direct and efficient framework. As an application, we construct new explicit families of LCP AG codes with determined parameters $[n,k,d]$, covering three ramification regimes. The resulting codes meet or approach the Goppa designed distance, offering greater flexibility for cryptographic applications.

math.AG↗

arXiv2025

A closed formula for the Geil-Matsumoto bound on numerical semigroups via Ap\'ery sets

The Geil-Matsumoto bound (GM bound) constrains the number of rational points on a curve over a finite field in terms of the Weierstrass semigroup of any of the points on the curve. For general numerical semigroups, the GM bound lacks a simple closed-form expression, making its computation a challenging problem. A closed formula has been obtained for the case when the semigroup is generated by two co-prime integers. In this work, for any numerical semigroup, we provide a closed formula for the GM bound in terms of the Ap\'ery set of a nonzero element of the semigroup. In the case where the numerical semigroup is generated by consecutive integers $n, n+1, \dots, n+t$ with $\lceil\textstyle\frac{n-1}{2}\rceil\leq t \leq n-1$, we obtain a simple closed formula for the bound. We apply these results to obtain upper bounds on the number of rational points for algebraic curves over finite fields. In some cases, our bounds improve some well-known upper bounds on the number of rational points.

math.NT↗

arXiv2025

Sequences of LCD AG codes and LCP of AG Codes attaining the Tsfasman-Vladut-Zink bound

Since Massey introduced linear complementary dual (LCD) codes in 1992 and Bhasin et al. later formalized linear complementary pairs (LCPs) of codes - structures with important cryptographic applications - these code families have attracted significant interest. We construct infinite sequences $(C_i)_{i \geq 1}$ of LCD codes and of LCPs $(C', D')_{i \geq 1}$ over $\mathbb{F}_{q^2}$ obtained from the Garcia-Stichtenoth tower of function fields, where we describe suitable non-special divisors of small degree on each level of the tower. These families attain the Tsfasman-Vl\u{a}du\c{t}-Zink bound and, for sufficiently large $q$ exceed the classic Gilbert-Varshamov bound, providing explicit asymptotically good constructions beyond existential results. We also exhibit infinite sequences of self-orthogonal over $\mathbb{F}_{q^2}$ and, when $q$ is even, self-dual codes from the same tower all meeting the Tsfasman-Vl\u{a}du\c{t}-Zink bound.

math.AG↗