ENG 
ESP Session S26 - Finite fields and applications
Talks
Thursday, July 15, 11:00 ~ 11:20 UTC-3
LCD codes arising from wavelets
Horacio Tapia-Recillas
Departamento de Matem\'aticas, Universidad Aut\'onoma Metropolitana-Iztapalapa, CDMX, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.
Since their introduction wavelets have provided a tool for handling several problems in science and engineering, including: audio denoising, signal compression, fingerprint compression, image recognition, etc. In this talk wavelet transforms will be defined over the finite field $\mathbb Z_p$ for some values of $p$, and LCD codes arising from these transforms are described.
Joint work with Fernanda D. de Melo Hern\'andez (Universidade Estadual de Maring\'a, PR, Brazil) and C\'esar A. Hern\'andez Melo (Universidade Estadual de Maring\'a, PR, Brazil).
Thursday, July 15, 11:30 ~ 11:50 UTC-3
On the classification of some Rational Cyclic AG Codes.
Gustavo Andrés Cabaña
Universidad Nacional del Litoral, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $F$ be the rational function field $\mathbb{F}_{q}(x)$ and $A=\mathrm{Aut}_{\mathbb{F}_{q}}(F)$. Consider $\sigma \in A$ and $P_{1},P_{2},\ldots ,P_{n},Q$ rational places of $F$ such that $\sigma(P_{i})=P_{i+1 \; \mathrm{mod} \; n}$ and $\sigma(Q)=Q$.
Let $D,G$ be divisors of $F$ given by $D=P_{1}+P_{2}+ \cdots + P_{n}$ and $G=rQ$ for some integer $r>0$.
We study AG codes $\mathcal{C_{L}}(D,G)$, which are cyclic AG codes, and we prove that, up to monomial equivalence, there is only one code in this family of a fixed length and dimension.
Joint work with María Chara (Universidad Nacional del Litoral, Argentina), Ricardo Podestá (Universidad Nacional de Córdoba, Argentina) and Ricardo Toledano (Universidad Nacional del Litoral, Argentina).
Thursday, July 15, 12:00 ~ 12:20 UTC-3
Bases of Riemann-Roch spaces from function fields of Kummer type and Algebraic Geometry Codes
Horacio Navarro
Universidad del Valle, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.
In 1981 Goppa defined the algebraic geometry codes (AG codes) as the image of Riemann-Roch spaces under the evaluation map at several rational places. It is important to note that the Riemann-Roch Theorem gives lower bounds for the dimension and minimum distance of this class of codes, however, to determine the dimension and a generator matrix it is necessary a basis of the associated Riemann-Roch space.
In 2005 Maharaj, Matthews and Pirsic constructed explicit bases of Riemann-Roch spaces related to the Hermitian function field based on its Kummer structure, in addition, they introduced the notion of the floor of a divisor in order to improve the bound of the minimum distance of the AG codes.
The purpose of this talk is to construct explicit bases of Riemann-Roch spaces associated to certain function fields of Kummer type, to calculate the floor of divisors with support on totally ramified places and finally to show examples of AG codes with good parameters.
Thursday, July 15, 12:30 ~ 12:50 UTC-3
Rational functions with Small Value Set
Luciane Quoos
Universidade Federal do Rio de Janeiro, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $q$ be a power of a prime $p$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For any rational function $h(x) \in \mathbb{F}_q(x)$, its value set is defined as $$ V_h =\{ h(\alpha) \mid \alpha \in \mathbb{P}^1(\mathbb{F}_q) \}\subset \mathbb{P}^1(\mathbb{F}_q)=\mathbb{F}_q \cup \{ \infty \} . $$ If $h(x) \in \mathbb{F}_q(x)$ is a rational function of degree $d$, then one has the trivial bound $$ \left \lceil \frac{q+1}{d} \right \rceil \leqslant \# V_h \leqslant q+1. $$
In connection with Galois Theory and Algebraic Curves, under certain hypothesis, it is proved that a function $h(x) \in \mathbb{F}_q(x)$ having a "small value set" is equivalent to the field extension $\mathbb{F}_q(x)/\mathbb{F}_q(h(x))$ being Galois.
Joint work with Daniele Bartoli (Università degli Studi di Perugia, Italy) and Herivelto Borges (Universidade de São Paulo, Brazil).
Thursday, July 15, 13:00 ~ 13:20 UTC-3
Constructions of APN permutations
Petr Lisonek
Simon Fraser University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let ${\mathbb F_q}$ denote the finite field with $q$ elements. Almost perfect nonlinear (APN) function is a mapping $f:{\mathbb F_{2^n}}\rightarrow{\mathbb F_{2^n}}$ such that the equation $f(x+a)+f(x)=b$ has at most two solutions for any fixed $a\in{\mathbb F_{2^n}^*}$ and $b\in{\mathbb F_{2^n}}$.
APN functions are of great interest in the design of cryptographic block ciphers as they provide the best resistance against differential cryptanalysis. Some block cipher designs require that the APN function is a permutation of ${\mathbb F_{2^n}}$. Several infinite families of APN permutations are known, but not much is known in general. The Big APN Problem asks whether the exist any APN permutations of ${\mathbb F_{2^n}}$ for even $n>6$.
In this talk we survey several recent results (our as well as due to other authors) on constructing APN permutations using vector spaces of Walsh zeros, as well as on the non-existence of APN permutations of certain specific forms (known as Kim functions).
Joint work with Benjamin Chase (Simon Fraser University, Canada).
Thursday, July 15, 14:00 ~ 14:20 UTC-3
Minimal value set polynomials and towers of Garcia, Stichtenoth and Thomas
Ricardo Toledano
Universidad Nacional del Litoral, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce the notions of $S_\phi$-polynomial and $S$-minimal value set polynomial where $\phi$ is a polynomial over a finite field $\mathbb{F}_q$ and $S$ is a finite subset of an algebraic closure of $\mathbb{F}_q$. We will show that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are $S_\phi$-minimal value set polynomials for $\phi=x^m$, whose $S$-value sets can be explicitly computed in terms of the monomial $x^m$. As a consequence of these results we will show that the tower recursively defined by the equation $y^2=2x(x-1)$ over $\mathbb{F}_q$ with $q=3^n\geq 9$, is the only one in the whole family having limit bigger than one.
Thursday, July 15, 14:30 ~ 14:50 UTC-3
Explicit Formulas for Polynomial Involutions Over Finite Fields
Ivelisse Rubio
University of Puerto Rico, Río Piedras, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Permutations of finite fields play an important role in communications as they are used in applications ranging from speech encryption to coding theory and cryptography. Involutions are permutations that are its own inverse and have the advantage that the same implementation can be used for coding and decoding. We present explicit formulas for all the involutions of finite fields that are given by monomials and for their fixed points. We also present partial results on explicit formulas that produce families of binomial involutions.
Joint work with Lillian González (University of Puerto Rico, Río Piedras) and Ariane Masuda (New York City College of Technology, The City University of New York (CUNY)).
Thursday, July 15, 15:00 ~ 15:20 UTC-3
Pairs of primitive elements on finite fields
Cícero Carvalho
Universidade Federal de Uberlândia, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we would like to present some results on the existence of pairs of elements in a finite field, where the first element is either primitive or primitive and normal over a subfield, and the second element is primitive and a rational function of the first one. We also will present some numerical results illustrating the main theoretical results.
Joint work with João Paulo Guardieiro (Universidade Federal de Uberlândia), Victor G.L. Neumann (Universidade Federal de Uberlândia) and Guilherme Tizziotti (Universidade Federal de Uberlândia).
Friday, July 16, 11:00 ~ 11:20 UTC-3
Clausen’s formula and high Picard rank K3 surfaces.
Adriana Salerno
Bates College, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Clausen’s formula is a classical identity characterizing certain hypergeometric series as squares of other hypergeometric series. Evans-Greene and Fuselier-Long-Ramakrishna-Swisher-Tu have described finite field analogues of this identity. Clausen’s formula also arises in the context of Picard-Fuchs equations satisfied by holomorphic forms on geometrically natural one-parameter families of K3 surfaces. We discuss the implications for point counting on such K3 surfaces over finite fields.
Joint work with Ursula Whitcher (Mathematical Reviews, AMS).
Friday, July 16, 11:30 ~ 11:50 UTC-3
On integral points on isotrivial elliptic curves over function fields
Ricardo Conceicao
Gettysburg College, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $k$ be a finite field and $L$ be the function field of a curve $C/k$. In this talk, we discuss certain arithmetical properties satisfied by integral points on elliptic curves over $L$ such that their $j$-invariant is an element of $k$. One particular result that we prove is that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of the size of $S$ and the genus of $C$.
Friday, July 16, 12:00 ~ 12:20 UTC-3
On the generalized Suzuki curve
Mariana Coutinho
Universidade de São Paulo, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $p$ be a prime number and, for $t>1$, consider $\mathcal{X}$ the nonsingular model of $$Y^{q}-Y= X^{q_0}(X^{q}- X),$$ where $q_{0}= p^{t}$ and $q=p^{2t-1}$.
For $p$ even, $\mathcal{X}$ is the so-called Deligne--Lusztig curve associated with the Suzuki group, which has remarkable properties, for instance its large automorphism group with respect to the genus.
In the present work, we address the study of $\mathcal{X}$ for $p$ an odd prime number.
Joint work with Herivelto Borges (Universidade de São Paulo, Brazil).
Friday, July 16, 12:30 ~ 12:50 UTC-3
Estimates on the number of rational solutions of variants of diagonal equations over finite fields
Mariana Valeria Pérez
Universidad Nacional de Hurlingham-CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
Several problems of coding theory, cryptography and combinatorics require the study of the set of $\mathbb{F}_{\hskip-0.7mm q}$-rational points (i.e. points with coordinates in the finite field of $q$ elements $\mathbb{F}_{\hskip-0.7mm q}$) of varieties defined over $\mathbb{F}_{\hskip-0.7mm q}$ on which the symmetric group of permutations of the coordinates acts (see, for example, [1], [2] and [3]).
In this work, we study the set of $\mathbb{F}_{\hskip-0.7mm q}$-rational solutions of equations defined by polynomials evaluated in power-sum polynomials with coefficients in $\mathbb{F}_{\hskip-0.7mm q}$. More precisely, we consider $m$th-power sum polynomials in the variables $X_1,\ldots,X_n$, namely, the polynomials of the form $P_m=X_1^m+\cdots+X_n^m.$ Let $f \in \mathbb{F}_{\hskip-0.7mm q}[Y_1,\ldots,Y_d]$ and $P_{m_1},\cdots,P_{m_d}\in \mathbb{F}_{\hskip-0.7mm q}[X_1,\ldots,X_n]$. We define the $\mathbb{F}_{\hskip-0.7mm q}$-affine hypersurface given by $f(P_{m_1}, \dots, P_{m_d}) + g$, where $g\in\mathbb{F}_{\hskip-0.7mm q}[X_1,\ldots,X_n]$. Under certain hypotheses on $f$ and $g$, we prove that this hypersurface is absolutely irreducible, and we obtain an upper bound of the dimension of its singular locus. These results are used to obtain estimates on the number of $\mathbb{F}_{\hskip-0.7mm q}$-rational points of this type of hypersurface by applying estimates for absolutely irreducible singular projective varieties provided in [4].
Finally we apply this methodology to the problem of estimating the number of $\mathbb{F}_{\hskip-0.7mm q}$-rational solutions of certain polynomial equations on $\mathbb{F}_{\hskip-0.7mm q}$. More precisely, we provide improved estimates and existence results of $\mathbb{F}_{\hskip-0.7mm q}$-rational solutions to the following equations: deformed diagonal equations, generalized Markoff-Hurwitz-type equations and Carlitz's equations (see, for example, [5]).
REFERENCES
[1] A. Cafure, G. Matera and M. Privitelli. Singularities of symmetric hypersurfaces and Reed-Solomon codes, Adv. Math. Commun. 6 (2012). no. 1, 69--94.
[2] E.Cesaratto, G. Matera , M. Pérez and Melina Privitelli. On the value set of small families of polynomials over a finite field. I. J. Combin. Theory Ser. A 124 (2014), 203--227.
[3] G. Matera, M.Pérez and M. Privitelli. Factorization patterns on nonlinear families of univariate polynomials over a finite field, J Algebr. Comb. (2019), 1-51.
[4] S. Ghorpade and G. Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J. 2 (2002), no. 3, 589--631.
[5] G. Mullen y D. Panario, Handbook of finite fields. CRC Press, Boca Raton, FL, 2013.
Joint work with Melina Privitelli (Universidad Nacional de General Sarmiento-CONICET, Argentina).
Friday, July 16, 14:00 ~ 14:20 UTC-3
On diagonal equations over finite fields via walks in NEPS of graphs
Denis Videla
Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
We obtain an explicit combinatorial formula for the number of solutions $(x_1,\ldots,x_r)\in \mathbb{F}_{p^{ab}}$ to the diagonal equation $x_{1}^k+\cdots+x_{r}^k=\alpha$ over the finite field $\mathbb{F}_{p^{ab}}$, with $k=\frac{p^{ab}-1}{b(p^a-1)}$ and $b>1$ by using the number of $r$-walks in NEPS of complete graphs. This talk is based on a recent accepted article.
\textsc{Denis E.\@ Videla}. \textit{On diagonal equations over finite fields via walks in NEPS of graphs}. Finite Fields Appl. (2021), accepted, https://arxiv.org/abs/1907.03145
Friday, July 16, 14:30 ~ 14:50 UTC-3
Finite Field Constructions of Ordered Covering Arrays
Lucia Moura
University of Ottawa, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Ordered covering arrays generalize both ordered orthogonal arrays and covering arrays, which are well-studied combinatorial designs. Classical codes using the Hamming metric can be generalized to codes with a poset metric. The Niederreiter-Rosenbloom-Tsfasman (NRT) metric corresponds to posets that are the disjoint union of chains of the same size. In this talk, we discuss finite field constructions of ordered covering arrays, and their use in upper bounds for NRT-metric covering codes.
Joint work with André Guerino Castoldi (Universidade Tecnológica Federal do Paraná, UTFPR Pato Branco, Brazil), Emerson Luiz do Monte Carmelo (Universidade Estadual de Maringá, Brazil), Daniel Panario (Carleton University, Canada) and Brett Stevens (Carleton University, Canada).
Friday, July 16, 15:00 ~ 15:20 UTC-3
Average-case complexity of the Euclidean algorithm with a fixed polynomial over a finite field
Guillermo Matera
Universidad Nacional de General Sarmiento and CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
Computing the greatest common divisor of two nonzero univariate polynomials with coefficients in a finite field $\mathbb{F}_q$ of $q$ elements is a critical task, which arises in connection with many problems of computational mathematics. The fundamental computational tool for this problem is the Euclidean algorithm, and many variants of it are known in the literature. In particular, its average-case complexity has been the subject of several papers by, e.g., J. von zur Gathen, B. Vallée and others. All these results consider the average, for fixed degrees $e>d>0$, over the set of pairs $(g,f)\in\mathbb{F}_q[T]\times \mathbb{F}_q[T]$ with $g$ monic of degree $e$ and $f$ either of degree at most $d$, or of degree less than $e$, assuming the uniform distribution of pairs. Nevertheless, there are important tasks which rely heavily on the computation of gcd's and lie outside the scope of these analyzes, as the standard algorithm for finding the roots in $\mathbb{F}_q$ of a polynomial $f\in \mathbb{F}_q[T]$ of degree less than $q$, which requires computing $\gcd(T^q-T,f)$.
In this talk we shall discuss the behavior of the Euclidean algorithm applied to pairs $(g,f)$ of elements of $\mathbb{F}_q[T]$ when the highest-degree polynomial $g$ is fixed. For this purpose, considering all the elements $f$ of fixed degree, we establish asymptotically optimal bounds in terms of $q$ for the number of elements $f$ which are relatively prime with $g$ and for the average degree of $\gcd(g,f)$. We also exhibit asymptotically optimal bounds for the average-case complexity of the Euclidean algorithm applied to pairs $(g,f)$ as above.
Joint work with Nardo Giménez (Universidad Nacional de General Sarmiento, Argentina), Mariana Pérez (Universidad Nacional de Hurlingham and CONICET, Argentina) and Melina Privitelli (Universidad Nacional de General Sarmiento and CONICET, Argentina).
Tuesday, July 20, 16:00 ~ 16:50 UTC-3
Nonvanishing for cubic $L$-functions over function fields
Matilde Lalín
Université de Montréal, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Chowla's conjecture predicts that $L (1/2, \chi)$ does not vanish for Dirichlet $L$-functions associated with primitive characters $\chi$. It was first conjectured for the case of $\chi$ quadratic. For that case, Soundararajan proved that at least 87.5\% of the values $L (1/2, \chi)$ do not vanish, by calculating the first mollified moments. For cubic characters, the first moment has been calculated by Baier and Young (on $\mathbb{Q}$), by Luo (for a restricted family on $\mathbb{Q} (\sqrt{-3})$), and on function fields by David, Florea, and Lalín. In this talk we prove that there is a positive proportion of cubic Dirichlet characters for which the corresponding $L$-function at the central value does not vanish in the function field case. We arrive at this result by computing the first mollified moment using techniques that we previously developed in our work on the first moment of cubic $L$-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwill, Harper, and Radziwill - Soundararajan. Our results are on function fields, but with additional work they could be extended to number fields, assuming the Generalized Riemann Hypothesis.
Joint work with Chantal David (Concordia University) and Alexandra Florea (Columbia University).
Tuesday, July 20, 17:00 ~ 17:50 UTC-3
Weights on $\mathbb{Z}/m\mathbb{Z}$ and the MacWilliams Identities
Jay A. Wood
Western Michigan University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
There are four well-known weights defined on $\mathbb{Z}/m\mathbb{Z}$: the Hamming weight, the Lee weight, the Euclidean weight, and the homogeneous weight. Each weight determines a weight enumerator for linear codes, counting the number of codewords whose weight is a given value. The MacWilliams identities give a relation between the Hamming weight enumerator of a linear code and that of its dual code. In contrast, the MacWilliams identities tend to fail for the other three weights. We will summarize what is known, with an emphasis on the homogeneous weight, where the MacWilliams identities fail for composite $m > 5$.
Tuesday, July 20, 19:00 ~ 19:50 UTC-3
Open Problems for Polynomials over Finite Fields
Daniel Panario
Carleton University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
We focus only on univariate polynomials over a finite field. We first comment on the existence and number of several classes of polynomials. Open problems are theoretical.
Then, we center in low-weight (irreducible) polynomials. The conjectures here are practically oriented.
Next, we focus on the dynamics of iterating functions over finite fields and their periodicity and permutational implications.
Finally, time allowing, we give brief descriptions of a selection of open problems from several classical areas of research including factorization of polynomials, special polynomials (permutations, PN/APN functions), and relations between integer numbers and polynomials.
Posters
Finite subgroups of $\operatorname{PGL}(2,K)$
Abraham Rojas
ICMC-USP, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study the finite subgroups of $\operatorname{PGL}(2,K)$ when $K$ is an algebraically closed field of positive characteristic. For this, we use the theory of Algebraic Function Fields in one variable. We show how this result can be applied to find the automorphism group of Artin-Schreier curves.\\
\textbf{Reference:} \\ VALENTINI, R.; MADAN, M. A Hauptsatz of L.E Dickson and Artin-Schreier extensions. Journal für die Reine und Angewandte Mathematik, n. 318, 1980.