Sean $S$ un anillo de polinomios sobre el campo $K$ e $I$ un ideal graduado de $S$. En este seminario definiremos dos funciones asociadas a $I$: la función de distancia mínima $\delta_{I}$ y la función huella ${fp}_{I}$. Para definir $\delta_{I}$ y ${fp}_{I}$ usamos la función de Hilbert, el grado (multiplicidad) y una base de Gröbner para $I$. Estudiamos estas funciones desde un punto de vista computacional usando métodos de bases de Gröbner e implementaciones en Macaulay $2$. También estudiamos estas funciones desde un punto de vista teórico y examinamos su comportamiento asintótico. Estas funciones pueden ser expresadas en términos de los invariantes algebraicos de $I$. Mostraremos que ${fp}_{I}$ es una cota inferior para $\delta_{I}$. Damos fórmulas para calcular ${fp}_{I}$ y $\delta_{I}$ en el caso de ciertas intersecciones completas. En el caso de ideales monomiales que son intersección completa $\delta_{I}$ es igual a ${fp}_{I}$ y exhibimos una fórmula explícita en términos de los grados de un conjunto minimal de generadores de $I$.

Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the code distance. This opens the possibility to use a permutation representation in metaheuristics to find the minimum distance of an arbitrary linear code, as an alternative to the classic discrete representation. The proposed model is polynomial time-dependent with respect to the dimension of the code and the size of the base finite field. It can be concluded from our experiments that usual limitations of discrete representation, such as high selective pressure, are palliated by means of the new representation. Our approach is able to find true minimum distances of general linear codes of medium size, and it is the first work, as far as we know, to address the problem for codes over finite fields with more than two elements using metaheuristics. Comparison with the existing methods in the literature suggests that our proposal is accurate, efficient and able to outperform previous approaches for BCH and EQR binary codes. As a by-product, we have been able to find some inaccuracies in the list of best known linear codes over the field with eight elements in the Magma Computational Algebra System.

We consider cyclic codes $C_\mathcal{L}$ associated to quadratic trace forms in $m$ variables $Q_R(x) = \mathrm{tr}_{q^m/q}(xR(x))$ determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over $\mathbb{F}_{q^m}$, and three related codes $C_{\mathcal{L},0}$, $C_{\mathcal{L},1}$ and $C_{\mathcal{L},2}$. We describe the spectra for all these codes when $\mathcal{L}$ is an even rank family, in terms of the distribution of ranks of the forms $Q_R$ in the family $\mathcal{L}$, and we also compute the complete weight enumerator for $C_\mathcal{L}$. In particular, considering the family $\mathcal{L} = \langle x^{q^\ell} \rangle$, with $\ell$ fixed in $\mathbb{N}$, we give the weight distribution of four parametrized families of cyclic codes $C_\ell$, $C_{\ell,0}$, $C_{\ell,1}$ and $C_{\ell,2}$ over ${F}_q$ with zeros $\{ \alpha^{-(q^\ell+1)} \}$, $\{ 1,\, \alpha^{-(q^\ell+1)} \}$, $\{ \alpha^{-1},\,\alpha^{-(q^\ell+1)} \}$ and $\{ 1,\,\alpha^{-1},\,\alpha^{-(q^\ell+1)}\}$ respectively, where $q = p^s$ with $p$ prime, $\alpha$ is a generator of ${F}_{q^m}^*$ and $m/(m,\ell)$ is even. Finally, we give simple necessary and sufficient conditions for Artin-Schreier curves $y^p-y = xR(x) + \beta x$, $p$ prime, associated to polynomials $R \in \mathcal{L}$ to be optimal. We then obtain several maximal and minimal such curves in the case $\mathcal{L} = \langle x^{p^\ell}\rangle$ and $\mathcal{L} = \langle x^{p^\ell}, x^{p^{3\ell}} \rangle$.

Let ${X}$ be a non-singular, projective, irreducible, algebraic curve of genus $g \geq 1$ over a finite field ${F}_{q}$. Fix $m$ distinct rational points $P_1,\ldots,P_m$ on ${X}$. The set $ H(P_{1}, \ldots , P_{m}) = \{(a_{1}, \ldots, a_{m}) \in \mathbb{N}_{0}^ {m} \mbox{ ; } \exists f \in {F}_q({X}) \mbox{ with } (f)_{\infty} = \sum_{i=1}^ {m} a_{i}P_{i} \} $ is called the Weierstrass Semigroup in the points $P_1,\ldots,P_m$. This semigroup is very important to calculate the parameters of algebraic geometry codes (AG codes) over ${X}$. In general, is very complicate determinate this semigroup and various efforts have been possible for certain types of curves. In 2018, joint with G. Tizziotti, we determinate the generator set $\Gamma(P_1,\ldots,P_m)$ of $H(P_1,\ldots,P_m)$ for curves ${X}$ with affine plane model $f(y)=g(x)$, using the concept of discrepancy on two rational points $P,Q$ over ${X}$, introduced by Duursma and Park. With certain conditions, we will show how we can calculate the set $\Gamma(P_1,\ldots,P_m)$ for two types of maximal curves which cannot covered by the Hermitian curve. The first family the curves that we present is the example given by Giulietti and Korchmáros: For $q=n^3$, with $n \geq 2$ a prime power, the $GK$ curve over ${F}_{q^2}$ is the curve in $\mathbb{P}^{3}(\overline{{F}}_{q^2})$ with equations $ Z^{n^2-n+1} = Y \sum_{i=0}^{n} (-1)^{i+1} X^{i(n-1)}, X^{n} + X = Y^{n+1}$. The second family was introduced in 2016, by Tafazolian, Teherán and Torres: For $a,b,s\geq 1,n\geq 3$ integers such that $n$ is odd. Let $q=p^a$ a power of a prime, $b$ is a divisor of $a$, $s$ is a divisor of $(q^n+1)/(q+1)$ and $c\in {F}_{q^2}$ with $c^{q-1}=-1$. We define the curve ${X}_{a,b,n,s}$ over ${F}_{q^{2n}}$ with equations $ cy^{q+1}=t(x):=\sum_{i=0}^{a/b-1}x^{p^{ib}}\;\;\mbox{and}\;\; z^{M}=y^{q^2}-y\;, \qquad\mbox{where }M=\frac{q^n+1}{s(q+1)}$.

In this talk we would like to say something about the works of Fernando Torres, and particularly about those works we did together. We would also like to share some reflections on him as a researcher colleague.

A simple way to define an evaluation code is as a linear code that depends on a set of functions and a set of points. The evaluation code is obtained when the functions are evaluated on the points. One of the most classical examples is a Reed-Solomon code. In this talk we will see how evaluation codes can be studied using basic tools of commutative algebra. Then we will focus on a family of evaluation codes known as Cartesian codes. We will study how they are defined and some of its applications to LCD codes, quantum codes, and locally recoverable codes.

The study of the complexity of solving a system of polynomial equations over a finite field is a central problem in symbolic computation, with a wide range of applications. In this talk, we focus on the applications to post-quantum cryptography, more specifically to multivariate cryptography. In multivariate cryptography, the security of the cryptosystem relies on the computational hardness of finding the solutions of a system of polynomial equations over a finite field. Therefore, estimating the complexity of solving a given system of polynomial equations produces an estimate for the security of the corresponding cryptosystem. Often, in their analysis, cryptographers make the assumption that the systems that they analyze are "random". In this talk, we discuss two distinct mathematical formulations for randomness of a system of equations. For each of them, we show estimates on the complexity of system solving by means of Groebner bases techniques. The new results that we present are joint work with M. Bigdeli, E. De Negri, M. Dizdarevic, R. Minko, and S. Tsakou.