Let V denote a vector space of finite positive dimension. An ordered triple of linear operators on V is said to be a Leonard triple whenever for each choice of element of the triple there exists a basis of V with respect to which the matrix representing the chosen element is diagonal and the matrices representing the other two elements are irreducible tridiagonal. A Leonard triple is said to be modular whenever for each choice of element there exists an antiautomorphism of End(V) which fixes the chosen element and swaps the other two elements. We study combinatorial structures associated with Leonard triples and modular Leonard triples. In the first part we construct a simplicial complex of Leonard triples. The simplicial complex of a Leonard triple is the smallest set of linear operators which contains the given Leonard triple with the property that if two elements of the set are part of a Leonard triple, then the third element of the triple is also in the set. In the second part we construct a Hamming association scheme from modular Leonard triples using a method used previously in the context of Grassmanian codes.

Let n be a positive integer. A Latin square of order n is an n×n array L such that each element of some n-set occurs in each row and in each column of L exactly once. It is well-known that one may construct a 4-class association scheme on the positions of a Latin square, where the relations are the identity, being in the same row, being in the same column, having the same entry, and everything else. We describe the subconstituent (Terwilliger) algebras of such an association scheme. One also may construct several strongly regular graphs on the positions of a Latin square, where adjacency corresponds to any subset of the nonidentity relations described above. We describe the local spectrum and subconstituent algebras of such strongly regular graphs. Finally, we study various notions of isomorphism for subconstituent algebras using Latin squares as examples.

Un code de permutations G(n,d) un sous-ensemble C de Sym(n) tel que la distance de Hamming D entre deux éléments de C est supérieure ou égale à d. Dans cette thèse, le groupe des isométries de (Sym(n),D) est déterminé et il est prouvé que ces isométries sont des automorphismes du schéma d'association induit sur Sym(n) par ses classes de conjugaison. Ceci mène, par programmation linéaire, à de nouveaux majorants de la taille maximale des G(n,d) pour n et d fixés et n compris entre 11 et 13. Des algorithmes de génération avec rejet d'objets isomorphes sont développés. Pour classer les G(n,d) non isométriques, des invariants ont été construits et leur efficacité étudiée. Tous les G(4,3) et les G(5,4) ont été engendrés à une isométrie près, il y en a respectivement 61 et 9445 (dont 139 sont maximaux et décrits explicitement). D’autres classes de G(n,d) sont étudiées. A permutation code G(n,d) is a subset C of Sym(n) such that the Hamming distance D between two elements of C is larger than or equal to d. In this thesis, we characterize the isometry group of the metric space (Sym(n),D) and we prove that these isometries are automorphisms of the association scheme induced on Sym(n) by the conjugacy classes. This leads, by linear programming, to new upper bounds for the maximal size of G(n,d) codes for n and d fixed and n between 11 and 13. We develop generating algorithms with rejection of isomorphic objects. In order to classify the G(n,d) codes up to isometry, we construct invariants and study their efficiency. We generate all G(4,3) and G(4,5)codes up to isometry; there are respectively 61 and 9445 of them. Precisely 139 out of the latter codes are maximal and explicitly described. We also study other classes of G(n,d)codes.

In this thesis, we will explore the structure of Terwilliger algebras over several different types of finite groups. We will begin by discussing what a Schur ring is, as well as providing many different results and examples of them. Following our discussion on Schur rings, we will move onto discussing association schemes as well as their properties. In particular, we will show every Schur ring gives rise to an association scheme. We will then define a Terwilliger algebra for any finite set, as well as discuss basic properties that hold for all Terwilliger algebras. After specializing to the case of Terwilliger algebras resulting from the orbits of a group, we will explore bounds of the dimension of such a Terwilliger algebra. We will also discuss the Wedderburn decomposition of a Terwilliger algebra resulting from the conjugacy classes of a group for any finite abelian group and any dihedral group.

<p>Un code de permutations G(n,d) un sous-ensemble C de Sym(n) tel que la distance de Hamming D entre deux éléments de C est supérieure ou égale à d. Dans cette thèse, le groupe des isométries de (Sym(n),D) est déterminé et il est prouvé que ces isométries sont des automorphismes du schéma d'association induit sur Sym(n) par ses classes de conjugaison. Ceci mène, par programmation linéaire, à de nouveaux majorants de la taille maximale des G(n,d) pour n et d fixés et n compris entre 11 et 13. Des algorithmes de génération avec rejet d'objets isomorphes sont développés. Pour classer les G(n,d) non isométriques, des invariants ont été construits et leur efficacité étudiée. Tous les G(4,3) et les G(5,4) ont été engendrés à une isométrie près, il y en a respectivement 61 et 9445 (dont 139 sont maximaux et décrits explicitement). D’autres classes de G(n,d) sont étudiées.<p><p><p><p> <p><p><p><p>A permutation code G(n,d) is a subset C of Sym(n) such that the Hamming distance D between two elements of C is larger than or equal to d. In this thesis, we characterize the isometry group of the metric space (Sym(n),D) and we prove that these isometries are automorphisms of the association scheme induced on Sym(n) by the conjugacy classes. This leads, by linear programming, to new upper bounds for the maximal size of G(n,d) codes for n and d fixed and n between 11 and 13. We develop generating algorithms with rejection of isomorphic objects. In order to classify the G(n,d) codes up to isometry, we construct invariants and study their efficiency. We generate all G(4,3) and G(4,5)codes up to isometry; there are respectively 61 and 9445 of them. Precisely 139 out of the latter codes are maximal and explicitly described. We also study other classes of G(n,d)codes.<p><p><p><p> / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished