Projects

Publications

• Some computational results concerning the spectrum of sets of latin squares. (2014). Rafael A. Arce-Nazario, Francis N. Castro, Javier Córdova, Kenneth Hicks, Gary L. Mullen, & Ivelisse M. Rubio.
  
  Abstract: We discuss some computational problems concerning the distribution of orthogonal pairs in sets of latin squares of small orders.
  
  Final version published in Quasigroups and Related Systems 22 (2014), 159-164.

• Some Enumerational Results Relating the Numbers of Latin and Frequency Squares of Order n. (2012). Francis N. Castro, Gary L. Mullen, & Ivelisse M. Rubio.
  
  Abstract: We discuss some enumerational results relating the numbers of F(n;λ1,…,λm) and F(n;λ′1,…,λ′k) frequency squares of order n. In particular, for any frequency vector (λ1, …, λm) of n, we discuss some enumerational results relating the number of F (n; λ1, …, λm) frequency squares and the number of latin squares of order n. In Section 4 we also discuss some enumerational results for latin rectangles.
  
  Final version published in AMS Contemporary Math, Volume 537 (2012), pp. 129-136 (May 2012).

Student Projects

Undergraduate Students

• 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.

• 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 $i$.

• 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.