Self-dual codes from 3-class association schemes

3-class association schemes are used to construct binary self-dual codes. We use the pure and bordered construction to get self-dual codes starting from the adjacency matrices of symmetric and non-symmetric 3-class association schemes. In some specific cases, we also study constructions of self-dual codes over Z k . For symmetric 3-class association schemes, we focus on the rectangular scheme and association schemes derived from symmetric designs

Codes over rings : maximum distance separability and self-duality /

Una parte imporante de la teoría de códigos es la de determinar cotas del número de palabras de un código. Uno de los problemas fundamentales de la teoría de códigos es encontrar códigos con la máxima distancia mínima d. Los investigadores han encontrado diferentes cotas superiores e inferiores para los códigos lineales y no lineales, por ejemplo cotas de Plotkin, Johnson, Singleton, Elias, Linear Programming, Griesmer, Gilbert y Varshamov. En esta tesis se ha estudiado la cota de Singleton, que es una cota superior de la distancia mínima de un código, y se han definido los códigos Z2Z4-aditivos a distancia máxima separable (MDS). Dos cotas diferentes se presentan en este trabajo en el que se han caracterizado todos los códigos Z2Z4-aditivos a distancia máxima separable con respecto a la cota de Singleton (MDSS) y condiciones en los parámetros para códigos Z2Z4-aditivos a distancia máxima separable con respecto a la cota obtenida a partir del rango (MDSR). La generación de nuevos códigos ha sido siempre un tema interesante, dando lugar al estudio de las propiedades de estos nuevos códigos generados y a establecer nuevos resultados. Los códigos autoduales son una clase importante de códigos. Hay numerosas construcciones de códigos autoduales a partir de objetos combinatorios. En este trabajo se han dado dos métodos para generar códigos autoduales a partir de esquemas de asociación de clase 3; las construcciones pure y bordered. Con estos dos métodos, se han obtenido códigos binarios autoduales a partir de esquemas de asociación de clase 3 no simétricos y códigos sobre Zk a partir de esquemas de asociación rectangulares. Borges, Dougherty y Fernández-Córdoba en 2011 presentaron un método para generar nuevos códigos Z2Z4-aditivos autoduales a partir de otros códigos Z2Z4-aditivos autoduales extendiendo su longitud. En este trabajo se ha comprobado si las propiedades como separabilidad, antipodalidad y el tipo del código se mantienen o no cuando se utiliza este método.Bounds on the size of a code are an important part of coding theory. One of the fundamental problems in coding theory is to find a code with largest possible distance d. Researchers have found different upper and lower bounds on the size of linear and nonlinear codes e.g., Plotkin, Johnson, Singleton, Elias, Linear Programming, Griesmer, Gilbert and Varshamov bounds. In this dissertation we have studied the Singleton bound, which is an upper bound on the minimum distance of a code, and have defined maximum distance separable (MDS) Z2Z4 additive codes. Two different forms of these bounds are presented in this work where we have characterized all maximum distance separable Z2Z4-additive codes with respect to the Singleton bound (MDSS) and strong conditions are given for maximum distance separable Z2Z4-additive codes with respect to the rank bound (MDSR). Generation of new codes has always been an interesting topic, where one can study the properties of these newly generated codes and establish new results. Self-dual codes are an important class of codes. There are numerous constructions of self-dual codes from combinatorial objects. In this work we have given two methods for generating self-dual codes from 3-class association schemes, namely pure construction and bordered construction. Binary self-dual codes are generated by using these two methods from non-symmetric 3-class association schemes and self-dual codes from rectangular association schemes are generated over Zk. Borges, Dougherty and Fernández-Córdoba in 2011 presented a method to generate new Z2Z4-additive self-dual codes from the existing Z2Z4-additive selfdual codes by extending their length. In this work we have verified whether properties like separability, antipodality and code Type are retained or not, when using this method

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201