arXiv subjects

Yuri da Silva

Publications and source records attributed to Yuri da Silva.

3 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 the Maximality, Weierstrass Semigroups, and Automorphism Group of the Curve $Y^{q+1} = X^n(X^n + 1)$

We study the algebraic curve over $\mathbb{F}_{q^2}$ defined by $y^{q+1} = x^n(x^n+1)$, where $n$ is a positive integer coprime to the characteristic. We first prove (when $q$ is odd) that the nonsingular model of this curve is $\mathbb{F}_{q^2}$-maximal if and only if $n \mid (q+1)$. Writing $n = \frac{q+1}{m}$, we obtain a family of maximal curves parameterized by the divisors $m$ of $q+1$, which extends the previously studied case $m=3$ corresponding to maximal curves with the third largest possible genus. For this family, we determine the Weierstrass semigroups at several classes of rational points, including those lying above the branch points of the natural projection. These semigroups are described explicitly in terms of $q$ and $m$, and exhibit different behaviors depending on the arithmetic properties of $m$. Moreover, we determine the full automorphism group of the curve under a mild condition on the characteristic. Our results extend an earlier work on the case $m=3$ and provide new insight into the structure of this family of maximal curves.

math.AG↗

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 $y^m = \prod_{i=1}^r (x-α_i)^{λ_i}$ 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↗