{"id":116385,"date":"2018-03-11T10:45:29","date_gmt":"2018-03-11T10:45:29","guid":{"rendered":"https:\/\/www.deberes.net\/tesis\/sin-categoria\/calculo-de-la-distancia-aparente-de-codigos-abelianos-codigos-bch-multivariables\/"},"modified":"2018-03-11T10:45:29","modified_gmt":"2018-03-11T10:45:29","slug":"calculo-de-la-distancia-aparente-de-codigos-abelianos-codigos-bch-multivariables","status":"publish","type":"post","link":"https:\/\/www.deberes.net\/tesis\/algebra\/calculo-de-la-distancia-aparente-de-codigos-abelianos-codigos-bch-multivariables\/","title":{"rendered":"C\u00e1lculo de la distancia aparente de c\u00f3digos abelianos. c\u00f3digos bch multivariables"},"content":{"rendered":"<h2>Tesis doctoral de <strong> Diana Haidive Bueno Carre\u00f1o <\/strong><\/h2>\n<p>Resumen  el objetivo principal de este trabajo es el estudio del c\u00e1lculo de la distancia aparente de un c\u00f3digo abeliano, la cual es una cota para su distancia m\u00ednima. Asimismo, tambi\u00e9n son objeto de este trabajo el desarrollo de la noci\u00f3n de cota bch y c\u00f3digo bch multivariable,  de las construcciones multivariables y  las aplicaciones a los c\u00f3digos c\u00edclicos que se desprenden de dichas nociones.   un primer problema concreto que abordamos en esta tesis es determinar cu\u00e1ndo en un c\u00f3digo c\u00edclico la mayor de sus cotas bch coincide con su distancia m\u00ednima. En este sentido hemos encontrado condiciones en t\u00e9rminos de los divisores del polinomio xr-1 en ciertas extensiones del cuerpo base de los c\u00f3digos que consideramos en este trabajo.  Es m\u00e1s, a partir de estos resultados mostramos un m\u00e9todo de construcci\u00f3n de c\u00f3digos para los cuales el m\u00e1ximo de sus cotas bch y su distancia m\u00ednima coinciden.  en 1970, p. Camion [1] extendi\u00f3 el estudio de la cota bch a la familia de los c\u00f3digos abelianos al introducir la noci\u00f3n de distancia aparente de un c\u00f3digo abeliano. En el caso de los c\u00f3digos c\u00edclicos, la distancia aparente y la (m\u00e1xima) cota bch del c\u00f3digo, coinciden. La distancia aparente de un c\u00f3digo abeliano en un anillo semisimple es el m\u00ednimo de la distancia aparente de ciertos polinomios que corresponden con la transformada de fourier discreta de los elementos de todos los subconjuntos del conjunto de idempotentes que pertenecen al c\u00f3digo. Esto implica que el c\u00e1lculo es de orden exponencial. As\u00ed, en la pr\u00e1ctica, el n\u00famero de operaciones requeridas es muy elevado, por lo que es pertinente plantearse la b\u00fasqueda de un m\u00e9todo alternativo que simplifique el original. En [2], r. E. Sabin realiz\u00f3 la primera reducci\u00f3n de los c\u00e1lculos para obtener la distancia aparente de un polinomio fijo usando ciertas manipulaciones de matrices en el contexto de los llamados &quot;2-d cyclic codes&quot; (c\u00f3digos abelianos en dos variables). A\u00fan cuando el m\u00e9todo de sabin simplifica el original, no ayuda en nada a reducir el n\u00famero de c\u00e1lculos necesarios en el c\u00f3mputo de la distancia aparente de un c\u00f3digo. As\u00ed que el problema de la complejidad sigui\u00f3 abierto.  en el trabajo de camion antes mencionado, puede comprobarse que la distancia aparente de un c\u00f3digo c\u00edclico es precisamente la distancia aparente de un polinomio asociado al idempotente generador del c\u00f3digo (concretamente, la transformada de fourier). Hay ejemplos que muestran que, en el caso multivariable, dicha igualdad no se verifica, as\u00ed que es natural preguntarse si en ese caso puede obtenerse la distancia aparente a partir de ciertas manipulaciones sobre dicho polinomio o espec\u00edficamente sobre la hipermatriz (de coeficientes) asociada a la imagen bajo la transformada de fourier del idempotente generador, respecto de ciertas ra\u00edces de la unidad prefijadas. \u00e9ste es el objetivo principal alcanzado en este trabajo: presentar un algoritmo para calcular la distancia aparente de un c\u00f3digo abeliano bas\u00e1ndose en el manejo de hipermatrices, de tal forma que la cantidad de operaciones involucradas se reduzca notablemente. De hecho, en el caso caso de dos variables se reduce hasta el orden lineal.     una vez que el algoritmo ha sido desarrollado, el siguiente paso natural ha sido presentar una noci\u00f3n de c\u00f3digo bch multivariable que nos ha permitido extender muchos de los resultados cl\u00e1sicos sobre c\u00f3digos bch c\u00edclicos. Adem\u00e1s, hemos encontrado aplicaciones de nuestras t\u00e9cnicas en la construcci\u00f3n de c\u00f3digos abelianos con distancia aparente predeterminada.    referencias  [1] p. Camion, abelian codes, mrc tech. Sum. Rep. # 1059, university of wisconsin, 1971. [2] r. Evans sabin,  on minimum distance bounds for abelian codes, applicable algebra in engineering communication and computing, springer-verlag, 1992.   abstract   the main goal of this work is the study of the computation of the apparent distance of an abelian code, which is a bound for its minimum distance. Other contribution of this work is the development of a notion of multivariate bch bound and multivariate bch code. We also present a way to construct these type of multivariate codes and some applications to cyclic codes which are derived from the mentioned  notions.   first, we deal with the problem of determining when the maximum of the bch bounds of a cyclic code equals its minimum distance. In this sense, we have found conditions in terms of the divisors of the polynomial  xr-1 in some field extensions of the ground field of the code.  Moreover,  from this results we show a method of construction of codes such that  the maximum of its  bch bounds coincide with its minimum distance.   in 1970, p. Camion [1] extended the study of the bch bound to the family of abelian codes by introducing  the notion of apparent distance of an abelian code. For any cyclic code,  its apparent distance is equal to the maximum of its  bch bounds.  The apparent distance of an abelian code, in a semisimple ring, is the minimum of the apparent distances of certain polynomials. These polynomials correspond to the discrete fourier transform of the elements of all subsets in the set of the idempotents in the code.  This implies that the computation has exponential order.  Consequently, it is pertinent to consider the search of an alternative method which simplifies the original one.   In [2], r. E. Sabin made the first reduction of the computations required to obtain the apparent distance of a fixed polynomial by using some matrix manipulations. Her work is applied to  &quot;2-d cyclic codes&quot; (abelian codes in two variables).  Even though sabin&apos;s method simplifies the original one, it does not reduce the number of computations needed to get the apparent distance of a code. So, the problem of complexity is still open.   in the mentioned paper of camion,  one may see that the apparent distance of a cyclic code equals the apparent distance of a polynomial associated to the idempotent generator of the code (specifically, its discrete fourier transform).  There are examples which show that in the multivariate case the equality does not hold. Therefore,  it is natural to wonder whether it is possible to obtain the apparent distance of a multivariate code from some manipulations on the (coefficients) hypermatrix  associated to the discrete fourier transform of its idempotent generator, with respect to certain fixed roots of unity.  This is the most important goal that we have reached in this work: to present an algorithm to compute the apparent distance of an abelian code, based on manipulations on hypermatrices, in such a way that the involved number of computations is reduced significantly.  In fact, in the two variables case it is reduced to the linear order.  once the algorithm has been developed, the next natural step has been to present a notion of multivariate bch code. This concept has allowed us to extend many of the classical results about cyclic bch codes. Moreover, we have found applications of our techniques to the construction of abelian codes with a fixed apparent distance. references [1] p. Camion, abelian codes, mrc tech. Sum. Rep. # 1059, university of wisconsin, 1971. [2] r. Evans sabin,  on minimum distance bounds for abelian codes, applicable algebra in engineering communication and computing, springer-verlag, 1992.<\/p>\n<p>&nbsp;<\/p>\n<h3>Datos acad\u00e9micos de la tesis doctoral \u00ab<strong>C\u00e1lculo de la distancia aparente de c\u00f3digos abelianos. c\u00f3digos bch multivariables<\/strong>\u00ab<\/h3>\n<ul>\n<li><strong>T\u00edtulo de la tesis:<\/strong>\u00a0 C\u00e1lculo de la distancia aparente de c\u00f3digos abelianos. c\u00f3digos bch multivariables <\/li>\n<li><strong>Autor:<\/strong>\u00a0 Diana Haidive Bueno Carre\u00f1o <\/li>\n<li><strong>Universidad:<\/strong>\u00a0 Murcia<\/li>\n<li><strong>Fecha de lectura de la tesis:<\/strong>\u00a0 26\/09\/2014<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3>Direcci\u00f3n y tribunal<\/h3>\n<ul>\n<li><strong>Director de la tesis<\/strong>\n<ul>\n<li>Juan  Jacobo Sim\u00f3n Pinero<\/li>\n<\/ul>\n<\/li>\n<li><strong>Tribunal<\/strong>\n<ul>\n<li>Presidente del tribunal: angel Del rio mateos <\/li>\n<li>joan josep Climent coloma (vocal)<\/li>\n<li>edgar Mart\u00ednez moro (vocal)<\/li>\n<li>  (vocal)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tesis doctoral de Diana Haidive Bueno Carre\u00f1o Resumen el objetivo principal de este trabajo es el estudio del c\u00e1lculo de 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