{"id":217,"date":"2010-11-05T14:21:46","date_gmt":"2010-11-05T14:21:46","guid":{"rendered":"http:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=217"},"modified":"2023-01-12T11:40:39","modified_gmt":"2023-01-12T11:40:39","slug":"projects","status":"publish","type":"page","link":"https:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=217","title":{"rendered":"Projects"},"content":{"rendered":"<div id=\"BalFunc\">\n<h2>Balanced Functions<\/h2>\n<\/div>\n<h3>Publications<\/h3>\n<ul>\n<li><a name=\"NFBSFGCLSC\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/2017-GeneralizationCusick-DesignsCodesCrypto.pdf\" target=\"_blank\"><em>New families of balanced symmetric functions and a generalization of Cusick, Li and Stanica&#8217;s conjecture<\/em>.<\/a> (2018). Rafael A. Arce-Nazario, Francis N. Castro, Oscar E. Gonz\u00e1lez, Luis A. Medina &amp; Ivelisse M. Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> 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.<\/em><\/p>\n<p><\/p>\n<p>Final version published in Designs, Codes and Cryptography, 86(3), 693-701, DOI 10.1007\/s10623-017-0351-7, 2018.<\/li>\n<\/ul>\n<h3>Student Projects<\/h3>\n<h4>Undergraduate Students<\/h4>\n<ul>\n<li><a name=\"NFBSFGCLSC\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/2017-GeneralizationCusick-DesignsCodesCrypto.pdf\" target=\"_blank\"><em>New families of balanced symmetric functions and a generalization of Cusick, Li and Stanica&#8217;s conjecture<\/em>.<\/a> (2018). Rafael A. Arce-Nazario, Francis N. Castro, Oscar E. Gonz\u00e1lez, Luis A. Medina &amp; Ivelisse M. Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> 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.<\/em><\/p>\n<p><\/p>\n<p>Final version published in Designs, Codes and Cryptography, 86(3), 693-701, DOI 10.1007\/s10623-017-0351-7, 2018.<\/li>\n<\/ul>\n<hr \/>\n<div id=\"TurCod\">\n<h2>Turbo Codes<\/h2>\n<\/div>\n<h3>Publications<\/h3>\n<ul>\n<li><a name = \"ACIUPM\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/11\/Algebraic-Construction-of-Interleavers-Using-Permutation-Monomials.pdf\" target=\"_blank\"><em>Algebraic Construction of Interleavers Using Permutation Monomials<\/em><\/a>. (2004). Carlos J. Corrada &amp; Ivelisse Rubio.<br \/>\n<br \/>\n<em><strong>Abstract<\/strong>: We present an algebraic construction for interleavers of length <em>p<sup>r<\/sup><\/em>, where <em>p<\/em> 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 <em>Z<sub>p<sup>r<\/sup><\/sub><\/em> using permutations of the elements of the finite field <em>F<sub>p<sup>r<\/sup><\/sub><\/em> given by monomials over the field.<\/em><br \/>\n<br \/>\nFinal version published IEEE International Conference on Communications (2004).<\/li>\n<\/ul>\n<ul>\n<li><a name = \"DITCRPSI\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/11\/Deterministic-Interleavers-for-Turbo-Codes-with-Random-like-Performance-and-Simple-Implementation.pdf\" target=\"_blank\"><em>Deterministic Interleavers for Turbo Codes with Random-like Performance and Simple Implementation<\/em><\/a>. (2003). Carlos J. Corrada &amp; Ivelisse Rubio.<br \/>\n<br \/>\n<em><strong>Abstract<\/strong>: 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.<\/em><br \/>\n<br \/>\nFinal version published 3rd International Symposium on Turbo Codes and Related Topics, (2003)<\/li>\n<\/ul>\n<h3>Student Projects<\/h3>\n<h4>Undergraduate Students<\/h4>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/LDPC-Paper.pdf\" target=\"_blank\">Low-Density Parity-Check Codes<\/a><\/em>. Poster. (2007). Jeranfer Berm\u00fadez, Richard Garc\u00eda &amp; Reynaldo L\u00f3pez.<br \/>\n<br \/>\n<a rel=\"noreferrer noopener\" href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/11\/Poster-LDPC2007final.pdf\" target=\"_blank\">[Poster]<\/a><br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Los c\u00f3digos correctores de errores se utilizan en la comunicaci\u00f3n digital para detectar y corregir errores en la transmisi\u00f3n o almacenamiento de la informaci\u00f3n. En esta investigaci\u00f3n estudiamos c\u00f3digos Low-Density Parity-Check (LDPC). Estos c\u00f3digos son generados por grafos bipartitos construidos con permutaciones de cuerpos finitos dadas por monomios. Nuestro prop\u00f3sito es encontrar construcciones que resulten en c\u00f3digos LDPC eficientes. Para esto estudiamos si existe relaci\u00f3n entre la descomposici\u00f3n c\u00edclica de la permutaci\u00f3n y el girth del grafo.<\/em><\/li>\n<\/ul>\n<hr \/>\n<div id=\"other\">\n<h2>Other<\/h2>\n<\/div>\n<h3>Publications<\/h3>\n<ul>\n<li><a name = \"EPC\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/2015-E-perfectCodes-BullInstCombina.pdf\" target=\"_blank\"><em>$E$-Perfect Codes<\/em>.<\/a> (2006). Francis N. Castro, Heeralal Janwa, Gary Mullen, &amp; Ivelisse Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong>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.<\/em><br \/>\n<br \/>\nFinal version published in Bull. of the Institute of Combinatorics and its Applications, Vol. 75, Sept. 2015, 83-90..\n<\/li>\n<\/ul>\n<ul>\n<li><a name = \"EAAKMDAQPBECC\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2010\/11\/G5-2006-ISIT-Revised.pdf\" target=\"_blank\"><em>An Elementary Approach to Ax-Katz, McEliece\u2019s Divisibility and Applications to Quasi-Perfect Binary 2-Error Correcting Codes<\/em>.<\/a> (2006). Francis N. Castro, Ivelisse Rubio, Hugues Randriam, Oscar Moreno &amp; H. F. Mattson, Jr.<br \/>\n<br \/>\n<em><strong>Abstract<\/strong>: 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 <em>F<\/em><sub>2<em>f<\/em><\/sub><em> <\/em> and minimum distance 5 are quasi-perfect for <em>f<\/em> \u2264 10. We also present elementary proofs of divisibility results that, in some cases, improve previous results.<\/em><br \/>\n<br \/>\nFinal version published in IEEE, ISIT (2006).\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Balanced Functions Publications New families of balanced symmetric functions and a generalization of Cusick, Li and Stanica&#8217;s conjecture. (2018). Rafael A. Arce-Nazario, Francis N. Castro, Oscar E. Gonz\u00e1lez, Luis A. Medina &amp; Ivelisse M. Rubio. Abstract: In general, the methods &hellip; <a href=\"https:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=217\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":32,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"onecolumn-page.php","meta":{"footnotes":""},"class_list":["post-217","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/217","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=217"}],"version-history":[{"count":90,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/217\/revisions"}],"predecessor-version":[{"id":2007,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/217\/revisions\/2007"}],"up":[{"embeddable":true,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/32"}],"wp:attachment":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=217"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}