Balanced Functions
Publications
- New families of balanced symmetric functions and a generalization of Cusick, Li and Stanica’s conjecture. (2018). 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 exponential 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.Final version published in Designs, Codes and Cryptography, 86(3), 693-701, DOI 10.1007/s10623-017-0351-7, 2018.
Student Projects
Undergraduate Students
- New families of balanced symmetric functions and a generalization of Cusick, Li and Stanica’s conjecture. (2018). 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 exponential 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.Final version published in Designs, Codes and Cryptography, 86(3), 693-701, DOI 10.1007/s10623-017-0351-7, 2018.
Turbo Codes
Publications
- Algebraic Construction of Interleavers Using Permutation Monomials. (2004). Carlos J. Corrada & Ivelisse Rubio.
Abstract: We present an algebraic construction for interleavers of length pr, where p is any prime. These interleavers are very simple to implement and have performance better than random interleavers and other known algebraic constructions. We construct a permutation of Zpr using permutations of the elements of the finite field Fpr given by monomials over the field.
Final version published IEEE International Conference on Communications (2004).
- Deterministic Interleavers for Turbo Codes with Random-like Performance and Simple Implementation. (2003). Carlos J. Corrada & Ivelisse Rubio.
Abstract: In this work we present a class of interleavers for turbo codes generated using monomials over finite fields that achieve as good or better performance than random interleavers while still very simple to implement. We have found a general case that outperforms the random interleaver for every size of the finite field tested. We also raise the question of which are the parameters needed to determine how good an interleaver is.
Final version published 3rd International Symposium on Turbo Codes and Related Topics, (2003)
Student Projects
Undergraduate Students
- 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.
Other
Publications
- $E$-Perfect Codes. (2006). Francis N. Castro, Heeralal Janwa, Gary Mullen, & Ivelisse Rubio.
Abstract:Perfect codes provide one of the most important and widely studied classes of error-correcting codes. The problem of classifying the parameters of perfect codes remained open until 1973. In this paper we generalize perfect codes to e-perfect codes over finite fields. We present a list of parameters for e-perfect codes and conjecture that the list is complete, provide constructions for almost all the e-perfect codes listed in the conjecture and present partial results on proving that there are no other parameters for e-perfect codes.
Final version published in Bull. of the Institute of Combinatorics and its Applications, Vol. 75, Sept. 2015, 83-90..
- An Elementary Approach to Ax-Katz, McEliece’s Divisibility and Applications to Quasi-Perfect Binary 2-Error Correcting Codes. (2006). Francis N. Castro, Ivelisse Rubio, Hugues Randriam, Oscar Moreno & H. F. Mattson, Jr.
Abstract: In this paper we present an algorithmic approach to the problem of the divisibility of the number of solutions to a system of polynomial equations. Using this method we prove that all binary cyclic codes with two zeros over F2f and minimum distance 5 are quasi-perfect for f ≤ 10. We also present elementary proofs of divisibility results that, in some cases, improve previous results.
Final version published in IEEE, ISIT (2006).