<p>Perfect codes and minimal distance of a code have great importance in the study of theoryof codes. The perfect codes are classified generally and in particular for the Lee metric.However, there are very few perfect codes in the Lee metric. The Lee metric hasnice properties because of its definition over the ring of integers residue modulo q. It isconjectured that there are no perfect codes in this metric for q > 3, where q is a primenumber.The minimal distance comes into play when it comes to detection and correction oferror patterns in a code. A few bounds on the number of codewords and minimal distanceof a code are discussed. Some examples for the codes are constructed and their minimaldistance is calculated. The bounds are illustrated with the help of the results obtained.</p>

Perfect codes and minimal distance of a code have great importance in the study of theoryof codes. The perfect codes are classified generally and in particular for the Lee metric.However, there are very few perfect codes in the Lee metric. The Lee metric hasnice properties because of its definition over the ring of integers residue modulo q. It isconjectured that there are no perfect codes in this metric for q &gt; 3, where q is a primenumber.The minimal distance comes into play when it comes to detection and correction oferror patterns in a code. A few bounds on the number of codewords and minimal distanceof a code are discussed. Some examples for the codes are constructed and their minimaldistance is calculated. The bounds are illustrated with the help of the results obtained.

El objetivo de esta tesis es definir códigos perfectos sobre diferentes espacios de señal multidimensionales. Para resolver este problema, esta memoria presenta una relación original entre las Teorías de Grafos, Números y Códigos. Uno de nuestros principales resultados es la propuesta de una métrica adecuada sobre constelaciones de señal de tipo cuadrático, hexagonal y cuatro-dimensional. Esta métrica es la distancia entre los vértices de una nueva clase de grafos de Cayley definidos sobre diferentes anillos de enteros, en concreto, los enteros de Gauss, Eisenstein-Jacobi y Lipschitz. Así, resolvemos el problema de Teoría de Grafos conocido como el cálculo del conjunto perfecto dominante sobre las familias de grafos definidas en esta memoria. Para cada caso, daremos una condición suficiente para obtener dicho conjunto. La obtención de estos conjuntos de dominación implica directamente la construcción de códigos perfectos sobre los alfabetos que se consideran.Además, se obtendrán algunos resultados de isomorfía y embebimiento de grafos. En particular, se establecerán las relaciones entre grafos circulantes, toroidales y los que se presentan en este trabajo. Más concretamente, se mostrará que siempre existen órdenes para los cuales un grafo Toro puede ser embebido en un grafo Gaussiano, de Esenstein-Jacobi o de Lipschitz. Esto implica que la conocida distancia de Lee es un caso particular de las métricas presentadas en este trabajo. / The aim of this work is to define perfect codes for different multidimensional signal spaces. To solve this problem, this thesis presents an original relationship among the fields of Graph Theory, Number Theory and Coding Theory. One of our main findings is the proposal of a suitable metric over quadratic, hexagonal and four-dimensional constellations of signal points. This metric is the distance among vertices of a new class of Cayley graphs defined over integer rings, namely Gaussian integers, the Eisenstein-Jacobi integers and the Lipschitz integers.A problem in Graph Theory known as the perfect dominating set calculation is solved over the families of graphs defined in this memory. A sufficient condition for obtaining such a set is given for each case. The obtention of these sets of domination directly yields to the construction of perfect codes for the alphabets under consideration. In addition, some isomorphism and graph embedding results are going to be obtained. Specially, the relations between circulant, toroidal and the graphs presented in this work are stated. In particular, there always exist orders for which a Torus graph can be embedded in Gaussian, Eisenstein-Jacobi and Lipschitz graphs. This implies that the well-known Lee distance is a subcase of the metrics presented in this research.