Student Research

Master Students

Jaziel Torres Fuentes (2020-2022)

Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays. (2022). Rafael Arce, Carlos Hernández, José Ortiz, Ivelisse Rubio, & Jaziel Torres.

Abstract: Linear complexity is an important parameter for arrays that are used in applications related to information security. In this work we present new results on the multidimensional linear complexity of periodic arrays obtained using the definition and method proposed in [2,6,11]. The results include a generalization of a bound for the linear complexity, a comparison with the measure of complexity for multisequences, and computations of the complexity of arrays with periods that are not relatively prime for which the “unfolding method” does not work. We also present conjectures for exact formulas and the asymptotic behavior of the complexity of some array constructions.

Multidimensional Costas Arrays and Periodic Properties. Thesis. (2022). Jaziel Torres.

Abstract: Costas arrays were first introduced for SONAR detection applications, but later became an interesting object of mathematical research. Several generalizations of Costas arrays to multiple dimensions have been proposed. In this thesis, we lay the ground for the study of multidimensional Costas arrays by proposing concepts and showing results that extend into the higher-dimensional realm most of what is known about the periodicity of two-dimensional Costas arrays. Among the most important results is, for large classes of arrays, the non-existence of multidimensional Costas arrays that preserve the Costas property in every commensurate window when extended periodically in all directions. We also extend to higher-dimensions the Golomb-Moreno Conjecture that asserts all circular Costas maps are those from the Welch construction, which had been proved in the two-dimensional case. We prove a weaker version of this conjecture in the higher-dimensional context.

Multidimensional Costas Arrays and Their Periodicity . (2022). Ivelisse Rubio, & Jaziel Torres.

Abstract: A novel higher-dimensional definition for Costas arrays is introduced. This defini- tion works for arbitrary dimensions and avoids some limitations of previous definitions. Some non-existence results are presented for multidimensional Costas arrays preserv- ing the Costas condition when the array is extended periodically throughout the whole space. In particular, it is shown that three-dimensional arrays with this property must have the least possible order; extending an analogous two-dimensional result by H. Taylor. Said result is conjectured to extend for Costas arrays of arbitrary dimensions.

Lillian González Albino (2019-2022)

Involutions of Finite Fields Obtained From Binomials of the Form $x^m(x^{\frac{q−1}{2}} + a)$. Thesis. (2022). Lillian González.

Abstract: Permutations of finite fields Fq have many applications ranging from cryptography and combinatorics to the theory of computation. In many of these applications, a permutation and its inverse are stored in memory. A good option to reduce the memory footprint is to generate the permutation with a polynomial at the time of implementation. A better option is to use a permutation polynomial that is its own inverse; in this case, the permutation is called an involution. In applications to cryptography, the number of fixed points is correlated with its cryptographic properties.
In 2018, Zheng et al. characterized involutions of the form $x^mh(x^s)$ over $F_q$, and, in 2017, Castro et al. gave explicit formulas for monomial involutions of $F_q$ and their fixed points. The next simplest polynomials to study would be the binomials.
&nbsp In this work we characterize involutions of $F_q$ of the form $x^m(x^{\frac{q−1}{2}} + a)$ in terms of the number of fixed points. We present explicit formulas for obtaining these involutions and formulas for their fixed points. Additionally, we give an improvement of Zheng et al.’s result for polynomial involutions of $F_q$ of the form $x^mh(x^{\frac{q−1}{2}})$.

Carlos E. Seda Damiani (2018-2020)

Solvability of systems of polynomial equations with multivariate polynomials as coefficients. Thesis. (2020). Carlos Seda.

Abstract: In [3] Castro, Moreno and Rubio generalize the results of Moreno-Moreno’s theorem that gives a bound for the power of a prime p to divide the number of common zeros of the multivariate polynomials $F_1, …, F_t$ over a finite field $F_q$. This generalization regarded the coefficients of the polynomials to be uni-variate polynomials over a finite field instead of plain elements of the finite field. The result led to improve a theorem of Carlitz, for the estimation of the number of variables needed so that a system of polynomial equations with coefficients in $F_q[X]$ can have non-trivial zeros. We generalize the results of Castro, Moreno and Rubio to polynomials whose coefficients are multivariate polynomials over finite fields.

Undergraduate Students

Carlos Hernández Franco (2020-2022)

Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays. (2022). Rafael Arce, Carlos Hernández, José Ortiz, Ivelisse Rubio, & Jaziel Torres.

Abstract: Linear complexity is an important parameter for arrays that are used in applications related to information security. In this work we present new results on the multidimensional linear complexity of periodic arrays obtained using the definition and method proposed in [2,6,11]. The results include a generalization of a bound for the linear complexity, a comparison with the measure of complexity for multisequences, and computations of the complexity of arrays with periods that are not relatively prime for which the “unfolding method” does not work. We also present conjectures for exact formulas and the asymptotic behavior of the complexity of some array constructions.

Javier I. Santiago Vélez (2019-2021)

Permutation binomials of the form x^r(x^{q – 1} + a) over F_q^e. (2022). Ariane Masuda, Ivelisse Rubio, & Javier Santiago.

Abstract: We present several existence and nonexistence results for permutation binomials of the form x^r(x^{q – 1} + a), where e ≥ 2 and a ∈ F^*_q^e. As a consequence, we obtain a complete characterization of such permutation binomials over F_q² , F_q³ , F_q⁴ , F_p⁵ , and F_p⁶ , where p is an odd prime.

Sums of Permutation Monomials. Technical Report. (2019). Karla Borges & Javier Santiago.

Abstract: Las permutaciones juegan un papel importante en las comunicaciones, ya que se utilizan en una variedad de aplicaciones, desde la teoría de códigos hasta la criptografía. Los polinomios que generan permutaciones se llaman polinomios de permutación. Nosotros estudiamos binomios de la forma $f(x) = x^m +Ax^n = x^n(x^{m−n} +A)$ sobre cuerpos finitos $F_p$, donde $x^m$ y $Ax^n$ son también monomios de permutación. El objetivo es encontrar las condiciones para que $f(x)$ permute $F_p$. Aquí demostramos que $f(x)$ nunca es polinomio de permutación para $F_p$, donde $q = 2p+1$, $p$ primo, y conjeturamos que si $x^n(x^{m−n} +A)$ es un polinomio de permutación, donde $m − n = \frac{p−1}{d}$ , entonces $A^d + (−1)^{d+1}$ es un residuo edésimo.

Andrés Ramos Rodríguez (2017-2021)>/p>

Differences of Functions with the Same Value Multiset. Presentation. (2021). Dylan Cruz & Andrés Ramos.

Abstract: In a recent article, Ullman and Velleman studied functions a from an abelian group $G$ to itself that can be expressed as a difference of two bijections $b$, $c$ from $G$ to itself. In this work we relax the condition that $b$ and $c$ are bijections and instead study functions that can be expressed as the difference of two functions with the same value multiset. We construct all possible functions $b$, $c$ with same value multiset, such that $a = b − c$. As a consequence, we obtain a stronger version of Hall’s theorem, which gives a description of $b$ and $c$ in terms of $a$. We conclude by presenting further directions and questions that relate this new approach to applications of Hall’s theorem.

Binomials that do not produce permutations. Poster. (2019). Dylan Cruz & Andrés Ramos.

Abstract: Permutations have important applications in fields such as coding theory and cryptography. We study binomials of the form $x^m(x^{\frac{q-3}{2} }+B)$ over finite fields $F_q$. In this research we prove that this class of binomials never permutes a finite fields when: (a) $A = \alpha^k$, where $k$ is odd and $alpha$ is a primitive root of the field, (b) $A = \alpha^k$, where $k$ is even, $\alpha$ is a primitive root of the field and $q = 4h+1$, $h ∈ N$. We are left with the case $A = \alpha^k$ where $k$ is even and $q = 4h + 3$, $h ∈ N$. We conjecture that the remaining class of polynomials do not permute $F_q$.

Involutions of Finite Fields Obtained from Binomials. Technical Report. (2019). Dylan Cruz & Andrés Ramos.

Abstract: Las permutaciones de cuerpos finitos tienen aplicaciones en varios campos de las Matemáticas y Ciencias de Cómputos, como lo es la Criptografía. En específico, las permutaciones que son su propia inversa (involuciones) son de interés porque ofrecen una ventaja al implementarlas ya que requieren menos memoria. Estudiamos binomios de la forma $x^m(x \frac{q−3}{2} +A)$ sobre$F_q$ con el propósito de encontrar las condiciones en $A$ y $q$ tal que el binomio sea una involución. Aquí probaremos que, si $m = 1$, esta familia de binomios nunca produce permutaciones cuando de $F_q$ donde $q$ es potencia de un primo impar. Conjeturamos que esto es cierto para toda $m$.

Dylan Cruz Fonseca (2017 – 2020)

Differences of Functions with the Same Value Multiset. Presentation. (2021). Dylan Cruz & Andrés Ramos.

Abstract: In a recent article, Ullman and Velleman studied functions a from an abelian group $G$ to itself that can be expressed as a difference of two bijections $b$, $c$ from $G$ to itself. In this work we relax the condition that $b$ and $c$ are bijections and instead study functions that can be expressed as the difference of two functions with the same value multiset. We construct all possible functions $b$, $c$ with same value multiset, such that $a = b − c$. As a consequence, we obtain a stronger version of Hall’s theorem, which gives a description of $b$ and $c$ in terms of $a$. We conclude by presenting further directions and questions that relate this new approach to applications of Hall’s theorem.

Binomials that do not produce permutations. Poster. (2019). Dylan Cruz & Andrés Ramos.

Abstract: Permutations have important applications in fields such as coding theory and cryptography. We study binomials of the form $x^m(x^{\frac{q-3}{2} }+B)$ over finite fields $F_q$. In this research we prove that this class of binomials never permutes a finite fields when: (a) $A = \alpha^k$, where $k$ is odd and $alpha$ is a primitive root of the field, (b) $A = \alpha^k$, where $k$ is even, $\alpha$ is a primitive root of the field and $q = 4h+1$, $h ∈ N$. We are left with the case $A = \alpha^k$ where $k$ is even and $q = 4h + 3$, $h ∈ N$. We conjecture that the remaining class of polynomials do not permute $F_q$.

Involutions of Finite Fields Obtained from Binomials. Technical Report. (2019). Dylan Cruz & Andrés Ramos.

Abstract: Las permutaciones de cuerpos finitos tienen aplicaciones en varios campos de las Matemáticas y Ciencias de Cómputos, como lo es la Criptografía. En específico, las permutaciones que son su propia inversa (involuciones) son de interés porque ofrecen una ventaja al implementarlas ya que requieren menos memoria. Estudiamos binomios de la forma $x^m(x \frac{q−3}{2} +A)$ sobre $F_q$ con el propósito de encontrar las condiciones en $A$ y $q$ tal que el binomio sea una involución. Aquí probaremos que, si $m = 1$, esta familia de binomios nunca produce permutaciones cuando de $F_q$ donde $q$ es potencia de un primo impar. Conjeturamos que esto es cierto para toda $m$.

Luis Quiñones (2019)

Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays. Presentation. (2019). Luis Quiñones & Jaziel Torres.

Abstract: Sequences and multidimensional periodic arrays with entries in finite fields have important applications in coding theory and cryptography. The correlations and the linear complexity of the sequences and multidimensional arrays are important parameters for many applications, especially those related to information security, and hardware implementation. The general goal of this research is to study different constructions of sequences and multidimensional periodic arrays and their correlation and complexity parameters. We give a proof for the exact value of the complexity of an array constructed using the composition method that was previously conjectured.

Jaziel Torres Fuentes (2019-2020)

Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays. Presentation. (2019). Luis Quiñones & Jaziel Torres.

Abstract: Sequences and multidimensional periodic arrays with entries in finite fields have important applications in coding theory and cryptography. The correlations and the linear complexity of the sequences and multidimensional arrays are important parameters for many applications, especially those related to information security, and hardware implementation. The general goal of this research is to study different constructions of sequences and multidimensional periodic arrays and their correlation and complexity parameters. We give a proof for the exact value of the complexity of an array constructed using the composition method that was previously conjectured.

Karla Borges (2019)

Sums of Permutation Monomials. Technical Report. (2019). Karla Borges & Javier Santiago.

Abstract: Las permutaciones juegan un papel importante en las comunicaciones, ya que se utilizan en una variedad de aplicaciones, desde la teoría de códigos hasta la criptografía. Los polinomios que generan permutaciones se llaman polinomios de permutación. Nosotros estudiamos binomios de la forma $f(x) = x^m +Ax^n = x^n(x^{m−n} +A)$ sobre cuerpos finitos $F_p$, donde $x^m$ y $Ax^n$ son también monomios de permutación. El objetivo es encontrar las condiciones para que $f(x)$ permute $F_p$. Aquí demostramos que $f(x)$ nunca es polinomio de permutación para $F_p$, donde $q = 2p+1$, $p$ primo, y conjeturamos que si $x^n(x^{m−n} +A)$ es un polinomio de permutación, donde $m − n = \frac{p−1}{d}$ , entonces $A^d + (−1)^{d+1}$ es un residuo edésimo.

Lillian González Albino (2015-2019)

Involutions of finite fields obtained from binomials . Technical Report. (2018). Lillian González.

Abstract: Las permutaciones sobre cuerpos finitos son importantes ya que tienen aplicaciones desde cifrados de voz hasta teoría de computabilidad y criptografía. Polinomios que generan permutaciones son llamados polinomios de permutación; si estos polinomios son su propia inversa, son llamados involuciones. En esta investigación, queremos caracterizar binomios de involución la forma $x^m(x^{\frac{q−1}{2}} + a)$ sobre cuerpos finitos de característica $p$.

Implementation of an algorithm to compute linear complexity of periodic arrays. Technical Report. (2016). Lillian González.

Abstract: Moreno y Terkel presentaron una construcción de arreglos periódicos multidimensionales con propiedades de buena correlación y complejidad. Para analizar la complejidad de estos arreglos se “desenvolvían” utilizando el Teorema del Residuo Chino y luego se analizaban con el algoritmo Berlekamp-Massey. Pero este método presentaba una restricción en las dimensiones del arreglo pues tenían que ser coprimos. En [4, 5] fue propuesto una teoría nueva para analizar la complejidad linear de arreglos multidimensionales que no tiene esta restricción; provee una definición de complejidad linear de arreglos multidimensionales que es consistente con la definición de complejidad linear en una dimensión. En esta investigación se implementa el algoritmo desarrollado en [3] sobre la teoría presentada en [4, 5].

Jeffry Matos (2015-2016)

Complexity of Multidimensional Periodic Arrays. Bachelor Thesis. (2016). Jeffry Matos.

Abstract: Para obtener arreglos que puedan tener aplicaciones en sistemas que utilizan marcas de agua digitales, criptografía o señales de radar multi-blanco, los mismos deben poseer buenas propiedades de correlación y complejidad [5]. Por esto es imperativo el poder tener una medida de la complejidad de un arreglo multidimensional que no presente limitaciones. Las bases de Gröbner [3] son conjuntos de polinomios que poseen propiedades algorítmicas muy ricas. Al utilizarlas, es posible generalizar la definiciónn de la medida de la complejidad lineal de una sucesión a un arreglo. Si calculamos alguna base de Gröebner que genere los polinomios asociados al arreglo periódico multidimensional, entonces podremos examinar la complejidad del arreglo porque esto permite obtener un conjunto $\Delta_{Val(A)}$ cuyo tamaño define la complejidad lineal del arreglo. En este trabajo se estudian algunas propiedades de arreglos construidos utilizando un método propuesto por Moreno y Tirkel [7]. Se utilizó el algoritmo de Rubio-Sweedler [9] para computar las bases de Gröbner del arreglo y con las mismas examinar la complejidad de arreglos periódicos multidimensionales obtenidos con construcciones en [7], comparar con la complejidad de los arreglos vistos como multisucesiones [8], formular conjeturas y obtener resultados.

Natalia Pacheco-Tallaj (2015-2016)

Explicit Formulas for Monomial Involutions over Finite Fields. (2017). Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj & Ivelisse Rubio.

Abstract: Permutations of finite fields have important applications in cryptography and coding theory. Involutions are permutations that are its own inverse and are of particular interest because the implementation used for coding can also be used for decoding. We present explicit formulas for all the involutions of Fq that are given by monomials and for their fixed points.

Dickson Polynomials that Generate Involutions with More Than 3 Fixed Points. Technical Report. (2016). Natalia Pacheco.

Abstract: Let $F_q$ be a finite field of prime order $q$. This research studies the Dickson Polynomials $D_i(x, a)$ that generate a permutation of $F_q$ such that $D_i$ decomposes into cycles of uniform length $2$, i.e. $D_i$ is an involution. Our goal is to provide a necessary and sufficient formula to construct $I$ such that the Dickson Polynomial $D_i(x, a)$ has a particular amount d of fixed points. We wish to find a formula for $I$ in terms of $d$ and $q$, similarly to what we were able to achieve with Permutation Monomials in [1].

Oscar E. González (2014-2016)

Exact divisibility of exponential sums associated to elementary symmetric Boolean functions. (2017). Oscar E. González, Raúl E. Negrón, Francis N. Castro, Luis A. Medina & Ivelisse M. Rubio.

[Poster]

Abstract:In this paper, we present an elementary method to compute or estimate the exact 2-divisibility of exponential sums associated to symmetric Boolean functions. As a direct consequence of these results, we prove some of the open cases of Cusick-Li-Stănică’s conjecture about balanced symmetric Boolean functions.

New families of balanced symmetric functions and a generalization of Cusick, Li and Stanica’s conjecture. (2015). Rafael A. Arce-Nazario, Francis N. Castro, Oscar E. González, Luis A. Medina & Ivelisse M. Rubio.

Abstract: In general, the methods to estimate the p-divisibility of exponen- tial sums or the number of solutions of systems of polynomial equations over finite fields are non-elementary. In this paper we present the covering method, an elementary combinatorial method that can be used to compute the exact p-divisibility of exponential sums over a prime field. The results here allow us to compute the exact p-divisibility of exponential sums of new families of polynomials, to unify and improve previously known results, and to construct families of systems of polynomial equations over finite fields that are solvable.

Raúl Negrón Otero (2014-2016)

Exact divisibility of exponential sums associated to elementary symmetric Boolean functions. (2017). Oscar E. González, Raúl E. Negrón, Francis N. Castro, Luis A. Medina & Ivelisse M. Rubio.

[Poster]

Abstract:In this paper, we present an elementary method to compute or estimate the exact 2-divisibility of exponential sums associated to symmetric Boolean functions. As a direct consequence of these results, we prove some of the open cases of Cusick-Li-Stănică’s conjecture about balanced symmetric Boolean functions.

Ramón L. Collazo (2013-2014)

Exact $p$-divisibility of exponential sums of polynomials over finite fields. Poster. (2015). Ramón L. Collazo, Julio J. del Cruz, Daniel E. Ramírez, Francis N. Castro & Ivelisse M. Rubio.

Abstract:An $n$-variable Boolean function F is a function defined over $F^n$ with values in $F$, the finite field with two elements. Our aim is to calculate the exact $2$ divisibility of exponential sums associated to Boolean functions. This allows us to determine whether a Boolean function is balanced, i.e. whether $|\{x \in F|f(x) = 1\}| = 2^{n−1}$, which can be difficult in general. We prove two theorems which give affirmative answers to some of the open cases of Cusick-Li-Stănică’s conjecture about the non-existence of certain balanced Boolean funtions.

Solvability of Systems of Polynomial Equations over Finite Fields. Technical Report. (2014). Ramón L. Collazo, Julio J. de la Cruz & Daniel E. Ramírez.

Abstract: An important problem in mathematics is to determine if a system of polynomial equations has or not solutions over a given set. We study systems of polynomial equations over finite fields $F_p$, $p$ prime, and look for sufficient conditions that guarantee their solvability over the field. Using the covering method of (Castro & Rubio, n.d.) we get conditions on the degrees of the terms that allow us to construct families of systems that have exact p-divisibility of the number of solutions and therefore guarantee the solvability of the system over the finite field.

Julio De La Cruz (2013-2014)

Exact $p$-divisibility of exponential sums of polynomials over finite fields. Poster. (2015). Ramón L. Collazo, Julio J. del Cruz, Daniel E. Ramírez, Francis N. Castro & Ivelisse M. Rubio.

Abstract:An $n$-variable Boolean function F is a function defined over $F^n$ with values in $F$, the finite field with two elements. Our aim is to calculate the exact $2$ divisibility of exponential sums associated to Boolean functions. This allows us to determine whether a Boolean function is balanced, i.e. whether $|\{x \in F|f(x) = 1\}| = 2^{n−1}$, which can be difficult in general. We prove two theorems which give affirmative answers to some of the open cases of Cusick-Li-Stănică’s conjecture about the non-existence of certain balanced Boolean funtions.

Solvability of Systems of Polynomial Equations over Finite Fields. Technical Report. (2014). Ramón L. Collazo, Julio J. de la Cruz & Daniel E. Ramírez.

Abstract: An important problem in mathematics is to determine if a system of polynomial equations has or not solutions over a given set. We study systems of polynomial equations over finite fields $F_p$, $p$ prime, and look for sufficient conditions that guarantee their solvability over the field. Using the covering method of (Castro & Rubio, n.d.) we get conditions on the degrees of the terms that allow us to construct families of systems that have exact p-divisibility of the number of solutions and therefore guarantee the solvability of the system over the finite field.

Daniel Ramirez Calderon (2013-2014)

Exact $p$-divisibility of exponential sums of polynomials over finite fields. Poster. (2015). Ramón L. Collazo, Julio J. del Cruz, Daniel E. Ramírez, Francis N. Castro & Ivelisse M. Rubio.

Abstract:An $n$-variable Boolean function F is a function defined over $F^n$ with values in $F$, the finite field with two elements. Our aim is to calculate the exact $2$ divisibility of exponential sums associated to Boolean functions. This allows us to determine whether a Boolean function is balanced, i.e. whether $|\{x \in F|f(x) = 1\}| = 2^{n−1}$, which can be difficult in general. We prove two theorems which give affirmative answers to some of the open cases of Cusick-Li-Stănică’s conjecture about the non-existence of certain balanced Boolean funtions.

Solvability of Systems of Polynomial Equations over Finite Fields. Technical Report. (2014). Ramón L. Collazo, Julio J. de la Cruz & Daniel E. Ramírez.

Abstract: An important problem in mathematics is to determine if a system of polynomial equations has or not solutions over a given set. We study systems of polynomial equations over finite fields $F_p$, $p$ prime, and look for sufficient conditions that guarantee their solvability over the field. Using the covering method of (Castro & Rubio, n.d.) we get conditions on the degrees of the terms that allow us to construct families of systems that have exact p-divisibility of the number of solutions and therefore guarantee the solvability of the system over the finite field.

Christian A. Rodríguez Encarnación (2011-2014)

Generating Permutation Trinomials over Finite Fields. Technical Report. (2014). Christian A. Rodríguez & Alex D. Santos.

Abstract: Permutation polynomials over finite fields have many applications in areas such as coding theory and cryptography. We consider polynomials of the form $F_{a,b} (X) = X(X^{\frac{q−1}{d_1}} + aX^{\frac{q−1}{d_2}} + b)$, where $a, b \in {F_q}^*$ and $d_1 < d_2$. We construct partitions of these polynomials where polynomials in the same partition have value sets of equal cardinality. As a consequence we provide families of permutation polynomials.

Number of Permutation Polynomials. Poster. (2014). Christian A. Rodríguez & Alex D. Santos.

Abstract: Given $q = p^r$, $d_1$ and $d_2$, we construct partitions of polynomials of the form $F_{a,b} (X) = X(X^{\frac{q−1}{d_1}} + aX^{\frac{q−1}{d_2}} + b)$, where $a, b \in {F_q}^*$, that have value sets of the same cardinality. As a consequence we provide families of permutation polynomials and of polynomials with small value sets.

Alex D. Santos Sosa (2011-2014)

Generating Permutation Trinomials over Finite Fields. Technical Report. (2014). Christian A. Rodríguez & Alex D. Santos.

Abstract: Permutation polynomials over finite fields have many applications in areas such as coding theory and cryptography. We consider polynomials of the form $F_{a,b} (X) = X(X^{\frac{q−1}{d_1}} + aX^{\frac{q−1}{d_2}} + b)$, where $a, b \in {F_q}^*$ and $d_1 < d_2$. We construct partitions of these polynomials where polynomials in the same partition have value sets of equal cardinality. As a consequence we provide families of permutation polynomials.

Number of Permutation Polynomials. Poster. (2014). Christian A. Rodríguez & Alex D. Santos.

Abstract: Given $q = p^r$, $d_1$ and $d_2$, we construct partitions of polynomials of the form $F_{a,b} (X) = X(X^{\frac{q−1}{d_1}} + aX^{\frac{q−1}{d_2}} + b)$, where $a, b \in {F_q}^*$, that have value sets of the same cardinality. As a consequence we provide families of permutation polynomials and of polynomials with small value sets.

Jean-Karlo Accetta (2010-2011)

Número de Waring en Cuerpos Finitos. Technical Report. (2011). Zahir Mejias & Jean-Karlo Accetta.

[Poster] [Slideshow]

Abstract: El número de Waring es el número mínimo de variables que necesita una ecuación de la forma x₁^d + ··· + x_n^d = β para que tenga solución en los números naturales para cualquier valor de la constante en los números naturales. Trabajamos en el estudio de generalizaciones de este problema cuando los valores de la constante y las soluciones se definen en cuerpos finitos. Hemos desarrollado un programa para calcular el número de Waring, y con el mismo hemos mejorado en resultados anteriormente publicados.

Zahir Mejias (2010-2011)

Número de Waring en Cuerpos Finitos. Technical Report. (2011). Zahir Mejias & Jean-Karlo Accetta.

[Poster] [Slideshow]

Abstract: El número de Waring es el número mínimo de variables que necesita una ecuación de la forma x₁^d + ··· + x_n^d = β para que tenga solución en los números naturales para cualquier valor de la constante en los números naturales. Trabajamos en el estudio de generalizaciones de este problema cuando los valores de la constante y las soluciones se definen en cuerpos finitos. Hemos desarrollado un programa para calcular el número de Waring, y con el mismo hemos mejorado en resultados anteriormente publicados.

Jeranfer Bermúdez (2007-2010)

Study of r-Orthogonality of Latin Squares. Poster. (2010). Jeranfer Bermúdez & Lourdes Morales.

[Presentation]

Abstract: A Latin square (LS) of order n, is an n × n array of n different elements, where in each row and each column the elements are never repeated. Latin squares have various applications in Coding Theory, Projective Geometry and others. Two Latin squares of order n are said to be r-orthogonal if when the squares are superimposed we get r distinct ordered pairs of symbols. We study generalizations of the r-orthogonality to sets of LSʼs. In this work we present preliminary results on some properties of these generalizations.

Some Properties of Latin Squares – Study of Mutually Orthogonal Latin Squares. Technical Report. (2009). Jeranfer Bermúdez & Lourdes Morales.

Abstract: A latin square of order n is an n x n matrix containing n distinct symbols (usually denoted by the non-negative integers from 0 to n-1) such that each symbol appears in each row and column exactly once. Latin squares have various applications in Coding Theory, Cryptography, Finite Geometries and in the design of statistical experiments, to name a few. Two latin squares of the same order are said to be orthogonal if, when superimposed, all the pairs that are formed are different. In our research we look for new constructions of mutually orthogonal latin squares (MOLS). We present some partial results and conjectures related to this.

Study of r-Orthogonality for Latin Squares. Technical Report. (2009). Jeranfer Bermúdez, Richard García & Reynaldo López.

Abstract: A Latin square (LS) of order n, is an n x n array of n different elements, where in each row and each column the elements are never repeated. Latin squares have various applications in Coding Theory and Cryptography. The famous Sudoku squares are examples of Latin squares. Two Latin squares of order n are said to be r-orthogonal if when the squares are superimposed we get r distinct ordered pairs of symbols. In this work we study generalizations of r-orthogonality to sets of LSs. Also, we will present some preliminary results on some of the properties of these generalizations.

Study of Latin Square Generating Polynomials. Technical Report. (2009). Jeranfer Bermúdez.

Abstract: A Latin Square of order n is an n x n matrix of n distinct elements (usually represented with the numbers from 0 to n-1), where each element appears in each row and in each column exactly once. Their various applications in Coding Theory, Cryptography and Processor Scheduling, just to mention a few, make Latin Squares a very interesting field to study. We our interested in patterns or tendencies that could relate Latin Squares, or sets of Latin Squares, or could give Maximum Orthogonality. For that reason we look for another way of constructing Latin Squares which we had not studied previously, using polynomials over finite fields.

Low-Density Parity-Check Codes. Poster. (2007). Jeranfer Bermúdez, Richard García & Reynaldo López.

[Poster]

Abstract: Los códigos correctores de errores se utilizan en la comunicación digital para detectar y corregir errores en la transmisión o almacenamiento de la información. En esta investigación estudiamos códigos Low-Density Parity-Check (LDPC). Estos códigos son generados por grafos bipartitos construidos con permutaciones de cuerpos finitos dadas por monomios. Nuestro propósito es encontrar construcciones que resulten en códigos LDPC eficientes. Para esto estudiamos si existe relación entre la descomposición cíclica de la permutación y el girth del grafo.

Lourdes Morales (2007-2010)

Study of r-Orthogonality of Latin Squares. Poster. (2010). Jeranfer Bermúdez & Lourdes Morales.

[Presentation]

Abstract: A Latin square (LS) of order n, is an n × n array of n different elements, where in each row and each column the elements are never repeated. Latin squares have various applications in Coding Theory, Projective Geometry and others. Two Latin squares of order n are said to be r-orthogonal if when the squares are superimposed we get r distinct ordered pairs of symbols. We study generalizations of the r-orthogonality to sets of LSʼs. In this work we present preliminary results on some properties of these generalizations.

Some Properties of Latin Squares – Study of Maximal Sets of Latin Squares. Technical Report. (2010). Lourdes Morales.

Abstract: A latin square of order n is an n x n matrix containing n distinct symbols (usually denoted by the non-negative integers from 0 to n-1) such that each symbol appears in each row and column exactly once. Latin squares have various applications in Coding Theory, Cryptography, Finite Geometries and in the design of statistical experiments, to name a few. Two latin squares of the same order are said to be r-orthogonal if you get r distinct ordered pairs when you superimpose them. In our research we look for new constructions of maximal sets of latin squares. We present some partial results and conjectures related to this.

Some Properties of Latin Squares – Study of Mutually Orthogonal Latin Squares. Technical Report. (2009). Jeranfer Bermúdez & Lourdes Morales.

Abstract: A latin square of order n is an n x n matrix containing n distinct symbols (usually denoted by the non-negative integers from 0 to n-1) such that each symbol appears in each row and column exactly once. Latin squares have various applications in Coding Theory, Cryptography, Finite Geometries and in the design of statistical experiments, to name a few. Two latin squares of the same order are said to be orthogonal if, when superimposed, all the pairs that are formed are different. In our research we look for new constructions of mutually orthogonal latin squares (MOLS). We present some partial results and conjectures related to this.

Armando Claudio Jiménez (2008-2009)

De Cuadrados de Frecuencia a Cuadrados Latinos. Slideshow. (2009). Armando Claudio.

Abstract: A Latin Square of order $n$ is an $n \times n$ matrix of $n$ distinct elements in which each element appears exactly once in each row and once in each column. These Latin Squares have various applications such as Coding Theory, Cryptography, Processor Scheduling, among others. By an $F(n; \lambda_1, \cdots , \lambda_m)$ frequency square is meant an $n × n$ array in which each of the numbers $i$ with $1 \leq i \leq m$ appears exactly $\lambda_i$ times in each row and each column. Let $(F; \lambda_1, \cdots , \lambda_m)$ be a frequency square of order $n$. For $i = 1, · · · , m$, by an $i − transversal$ is meant a set of $n$ cells, one in each row and one in each column, each containing the symbol $i$.

Richard García Lebrón (2008-2009)

Study of r-Orthogonality for Latin Squares. Technical Report. (2009). Jeranfer Bermúdez, Richard García & Reynaldo López.

Abstract: A Latin square (LS) of order n, is an n x n array of n different elements, where in each row and each column the elements are never repeated. Latin squares have various applications in Coding Theory and Cryptography. The famous Sudoku squares are examples of Latin squares. Two Latin squares of order n are said to be r-orthogonal if when the squares are superimposed we get r distinct ordered pairs of symbols. In this work we study generalizations of r-orthogonality to sets of LSs. Also, we will present some preliminary results on some of the properties of these generalizations.

Low-Density Parity-Check Codes. Poster. (2007). Jeranfer Bermúdez, Richard García & Reynaldo López.

[Poster]

Abstract: Los códigos correctores de errores se utilizan en la comunicación digital para detectar y corregir errores en la transmisión o almacenamiento de la información. En esta investigación estudiamos códigos Low-Density Parity-Check (LDPC). Estos códigos son generados por grafos bipartitos construidos con permutaciones de cuerpos finitos dadas por monomios. Nuestro propósito es encontrar construcciones que resulten en códigos LDPC eficientes. Para esto estudiamos si existe relación entre la descomposición cíclica de la permutación y el girth del grafo.

Reynaldo López (2007-2009)

Low-Density Parity-Check Codes. Poster. (2007). Jeranfer Bermúdez, Richard García & Reynaldo López.

[Poster]

Abstract: Los códigos correctores de errores se utilizan en la comunicación digital para detectar y corregir errores en la transmisión o almacenamiento de la información. En esta investigación estudiamos códigos Low-Density Parity-Check (LDPC). Estos códigos son generados por grafos bipartitos construidos con permutaciones de cuerpos finitos dadas por monomios. Nuestro propósito es encontrar construcciones que resulten en códigos LDPC eficientes. Para esto estudiamos si existe relación entre la descomposición cíclica de la permutación y el girth del grafo.