{"id":1052,"date":"2022-06-07T02:17:34","date_gmt":"2022-06-07T02:17:34","guid":{"rendered":"http:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=1052"},"modified":"2023-01-11T00:58:49","modified_gmt":"2023-01-11T00:58:49","slug":"projects","status":"publish","type":"page","link":"https:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=1052","title":{"rendered":"Projects"},"content":{"rendered":"<div id=\"CostArr\">\n<h2>Costas Arrays<\/h2>\n<\/div>\n<h3>Publications<\/h3>\n<ul>\n<li><a name=\"CCMMACCS\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/2022-CircularCostasPublicadoArXiv.pdf\" target=\"_blank\"><em>Circular Costas maps: a multidimensional analog of circular Costas sequences <\/em><\/a>. (2022). Ivelisse Rubio, &amp; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A unifying theoretical framework is presented, in which the connections among Costas sequences, circular Costas sequences, Costas polynomials, the shifting property, and Welch sequences are extended to the multidimensional context. Several conjectures on multidimensional periodic Costas arrays by J. Ortiz-Ubarri et al. are proved. Furthermore, a conjecture on Costas polynomials over extension fields presented by Muratovi\u0107-Ribi\u0107 et al. is showed to be a multidimensional extension of a conjecture by Golomb and Moreno on circular Costas sequences. A weaker version of said conjecture is proved by considering a multidimensional extension of the shifting Costas property defined by O. Moreno.<\/em><br \/>\n<br \/>\nPublished in https:\/\/doi.org\/10.48550\/arXiv.2210.16661.\n<\/li>\n<\/ul>\n<ul>\n<li><a name=\"MCATP\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/10\/2022-MDCostas-ArXiv.pdf\" target=\"_blank\"><em>Multidimensional Costas Arrays and Their Periodicity <\/em><\/a>. (2022). Ivelisse Rubio, &amp; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> 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.<\/em><\/p>\n<p><\/p>\n<p>Published in https:\/\/doi.org\/10.48550\/arXiv.2208.02378.<\/li>\n<\/ul>\n<h3>Student Projects<\/h3>\n<h4> Master Students<\/h4>\n<ul>\n<li><a name=\"CCMMACCS\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/2022-CircularCostasPublicadoArXiv.pdf\" target=\"_blank\"><em>Circular Costas maps: a multidimensional analog of circular Costas sequences <\/em><\/a>. (2022). Ivelisse Rubio, &amp; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> A unifying theoretical framework is presented, in which the connections among Costas sequences, circular Costas sequences, Costas polynomials, the shifting property, and Welch sequences are extended to the multidimensional context. Several conjectures on multidimensional periodic Costas arrays by J. Ortiz-Ubarri et al. are proved. Furthermore, a conjecture on Costas polynomials over extension fields presented by Muratovi\u0107-Ribi\u0107 et al. is showed to be a multidimensional extension of a conjecture by Golomb and Moreno on circular Costas sequences. A weaker version of said conjecture is proved by considering a multidimensional extension of the shifting Costas property defined by O. Moreno.<\/em><br \/>\n<br \/>\nPublished in https:\/\/doi.org\/10.48550\/arXiv.2210.16661.\n<\/li>\n<\/ul>\n<ul>\n<li><a name=\"MCATP\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/10\/2022-MDCostas-ArXiv.pdf\" target=\"_blank\"><em>Multidimensional Costas Arrays and Their Periodicity <\/em><\/a>. (2022). Ivelisse Rubio, &amp; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> 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.<\/em><\/p>\n<p><\/p>\n<p>Published in https:\/\/doi.org\/10.48550\/arXiv.2208.02378.<\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2023\/01\/TesisJaziel-Multidimensional_Costas-Mayo2022.pdf.zip\" target=\"_blank\">Multidimensional Costas Arrays and Periodic Properties<\/a><\/em>. Thesis. (2022). Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> 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. <\/em><\/li>\n<\/ul>\n<hr \/>\n<div id=\"MultLinCom\">\n<h2>Multidimensional Linear Complexity<\/h2>\n<\/div>\n<h3>Publications<\/h3>\n<ul>\n<li><a name=\"ACMLCPA\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/10\/2022-AnalysisComputationMDLinearComplexity-ArXiv.pdf\" target=\"_blank\"><em>Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays<\/em><\/a>. (2022). Rafael Arce, Carlos Hern\u00e1ndez, Jos\u00e9 Ortiz, Ivelisse Rubio, &amp; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Linear complexity is an important parameter for arrays that are used in applications related to information security. \u00a0In 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 &#8220;unfolding method&#8221; does not work. We also present conjectures for exact formulas and the asymptotic behavior of the complexity of some array constructions. <\/em><\/p>\n<p><\/p>\n<p>Published in https:\/\/doi.org\/10.48550\/arXiv.2207.14398.<\/li>\n<\/ul>\n<ul>\n<li><a name=\"LCAMPA\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/05\/ComplexityNoFormated.pdf\" target=\"_blank\"><em>Linear complexity analysis of multidimensional periodic arrays<\/em><\/a>. (2019). Rafael Arce-Nazario, Francis Castro, Domingo G\u00f3mez-P\u00e9rez, Oscar Moreno, Jos\u00e9 Ortiz-Ubarri, Ivelisse Rubio, &amp; Andrew Tirkel.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> The linear complexity of a sequence is an important parameter for many applications, especially those related to information security, and hardware implementation. It is desirable to develop a corresponding measure and theory for multidimensional arrays that are consistent with those of sequences. In this paper we use Gr\u00f6bner bases to develop a theory for analyzing the multidimensional linear complexity of general periodic arrays. We also analyze arrays constructed using the method of composition and establish tight bounds for their multidimensional linear complexity.<\/em><br \/>\n<br \/>\nFinal version published in Applicable Algebra in Engineering, Communication and Computing, 1-21, doi: 10.1007\/s00200-019-00393-z, 2019.<\/li>\n<\/ul>\n<ul>\n<li><a name=\"FGBIRRMDPA\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/2016-GBforPeriodicArraysFq.pdf\" target=\"_blank\"><em>Finding a Gr\u00f6bner basis for the ideal of recurrence relations on $m$-dimensional periodic arrays<\/em><\/a>. (2016). Chris Heegard, Ivelisse Rubio, &amp; Moss Sweedler.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Recent developments in applications of multidimensional periodic arrays [9] have drawn new attention to the computation of Gr\u00f6bner bases for the ideal of linear recurrence relations on the arrays. An m-dimensional infinite array can be represented by a multivariate power series sitting within the ring of multivariate Laurent series. We reinterpret the problem of finding linear recurrence relations on m-dimensional periodic arrays as finding the kernel of a module map involving quotients of Laurent series and present an algorithm to compute a Gr\u00f6bner basis for this kernel. The algorithm does not assume the knowledge of a generating set for the kernel of this ideal and it is based on linear algebra computations. Finding a generating set is one application of the algorithm<\/em><br \/>\n<br \/>\nFinal version published in Contemporary Developments in Finite Fields and Applications (pp.296-320). Worls Scientific, June, 2016.<\/li>\n<\/ul>\n<ul>\n<li><a name=\"LCMANI\"><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/12\/2015-LinearComplexityISIT.pdf\" target=\"_blank\"><em>Linear Complexity for Multidimensional Arrays &#8211; a Numerical Invariant<\/em><\/a>. (2015). Tom H\u00f8holdt, Domingo G\u00f3mez-P\u00e9rez, Oscar Moreno, &amp; Ivelisse Rubio.<br \/>\n<br \/>\n<em><strong>Abstract:<\/strong> Linear complexity is a measure of how complex a one dimensional sequence can be. In this paper we extend the concept of linear complexity to multiple dimensions and present a definition that is invariant under well-orderings of the arrays. As a result we find that our new definition for the process introduced in the patent titled \u201cDigital Watermarking\u201d produces arrays with good asymptotic properties.<\/em><br \/>\n<br \/>\nFinal version published in Proceedings of the IEEE International Symposium on Information Theory (ISIT 2015). IEEE, 2015. p. 2697-2701.<\/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\/2022\/09\/Torrres_Presentation_Multi_LinearComplexity.pdf\" target=\"_blank\">Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays<\/a><\/em>. Presentation. (2019). Luis Qui\u00f1ones &#038; Jaziel Torres.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>  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.<\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/09\/TesinaJeff.pdf\" target=\"_blank\">Complejidad de arreglos peri\u00f3dicos multidimensionales<\/a><\/em>. Bachelor Thesis. (2016). Jeffry Matos.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong> Para obtener arreglos que puedan tener aplicaciones en sistemas que utilizan marcas de agua digitales, criptograf\u00eda o se\u00f1ales de radar multi-blanco, los mismos deben poseer buenas propiedades de correlaci\u00f3n 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 Gr\u00f6bner [3] son conjuntos de polinomios que poseen propiedades algor\u00edtmicas muy ricas. Al utilizarlas, es posible generalizar la definici\u00f3n de la medida de la complejidad lineal de una sucesi\u00f3n a un arreglo. Si calculamos alguna base de Gr\u00f6ebner que genere los polinomios asociados al arreglo peri\u00f3dico multidimensional, entonces podremos examinar la complejidad del arreglo porque esto permite obtener un conjunto $\\Delta_{Val(A)}$ cuyo tama\u00f1o define la complejidad lineal del arreglo. En este trabajo se estudian algunas propiedades de arreglos construidos utilizando un m\u00e9todo propuesto por Moreno y Tirkel [7]. Se utiliz\u00f3 el algoritmo de Rubio-Sweedler [9] para computar las bases de Gr\u00f6bner del arreglo y con las mismas examinar la complejidad de arreglos peri\u00f3dicos multidimensionales obtenidos con construcciones en [7], comparar con la complejidad de los arreglos vistos como multisucesiones [8], formular conjeturas y obtener resultados.  <\/em><\/li>\n<\/ul>\n<ul>\n<li><em><a href=\"http:\/\/ccom.uprrp.edu\/~labemmy\/Wordpress\/wp-content\/uploads\/2022\/11\/LillianGonzalez-ImplemAlgoriCompLineal-Mayo2016.pdf\" target=\"_blank\">Algoritmo para computar la complejidad linear de arreglos periodicos multidimensionales<\/a><\/em>. Technical Report. (2016). Lillian Gonz\u00e1lez.<br \/>\n<br \/>\n<em><strong>Abstract: <\/strong>  Moreno y Terkel presentaron una construcci\u00f3n de arreglos peri\u00f3dicos multidimensionales con propiedades de buena correlaci\u00f3n y complejidad. Para<br \/>\nanalizar la complejidad de estos arreglos se \u201cdesenvolv\u00edan&#8221; utilizando el Teorema del Residuo Chino y luego se analizaban con el algoritmo Berlekamp-Massey. Pero este m\u00e9todo presentaba una restricci\u00f3n en las dimensiones del arreglo pues ten\u00edan que ser coprimos. En [4, 5] fue propuesto una teor\u00eda nueva para analizar la<br \/>\ncomplejidad linear de arreglos multidimensionales que no tiene esta restricci\u00f3n; provee una definici\u00f3n de complejidad linear de arreglos multidimensionales que es consistente con la definici\u00f3n de complejidad linear en una dimensi\u00f3n. En esta investigaci\u00f3n se implementa el algoritmo desarrollado en [3] sobre la teor\u00eda presentada en [4, 5].<\/em><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Costas Arrays Publications Circular Costas maps: a multidimensional analog of circular Costas sequences . (2022). Ivelisse Rubio, &amp; Jaziel Torres. Abstract: A unifying theoretical framework is presented, in which the connections among Costas sequences, circular Costas sequences, Costas polynomials, the &hellip; <a href=\"https:\/\/ccom.uprrp.edu\/~labemmy\/?page_id=1052\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":1041,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"onecolumn-page.php","meta":{"footnotes":""},"class_list":["post-1052","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/1052","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=1052"}],"version-history":[{"count":67,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/1052\/revisions"}],"predecessor-version":[{"id":1998,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/1052\/revisions\/1998"}],"up":[{"embeddable":true,"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=\/wp\/v2\/pages\/1041"}],"wp:attachment":[{"href":"https:\/\/ccom.uprrp.edu\/~labemmy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1052"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}