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# Association schemes, codes, and difference sets

## Citation

Salazar-Lazaro, Carlos Harold (2007) Association schemes, codes, and difference sets. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-05222007-003651

## Abstract

This thesis consists of an introductory chapter and three independent chapters. In the first chapter, we give a brief description of the three independent chapters: abelian \$n\$-adic codes; generalized skew hadamard difference sets; and equivariant incidence structures. In the second chapter, we introduce \$n\$-adic codes, a generalization of the Duadic Codes studied by Pless and Rushanan, and we solve the corresponding existence problem. We introduce \$n\$-adic groups, canonical pplitters, and Margarita Codes to generalize the "self-dual" codes of Rushanan and Pless, and we solve the corresponding existence problem. In the third chapter, we consider the generalized skew hadamard difference set (GSHDS) existence problem. We introduce the combinatorial matrices \$A_{G,G_1}\$, where \$G_1=(Z/ exp(G)Z)^*\$ and \$G\$ is a group, to reduce the existence problem to an integral equation. Using a special finite-dimensional algebra or Association Scheme, we show \$A_{G,G_1}^2 = frac{|G|}{p}I\$ for general \$G\$. With the aid of \$A_{G,G_1}\$, we show some necessary conditions for the existence of a GSHDS \$D\$ in the group \$(Z/ p Z) times (Z/ p^2Z)^{2alpha+1}\$, we provide a proof of Johnsen's exponent bound, we provide a proof of Xiang's exponent bound, and we show a necessary existence condition for general \$G\$. In the fourth chapter, we study the incidence matrices \$W_{t,k}(v)\$ of \$t\$-subsets of \${1,ldots,v}\$ vs. \$k\$-subsets of \${1,ldots,v}\$. Also, given a group \$G\$ acting on \${1,ldots,v}\$, we define analogous incidence matrices \$M_{t,k}\$ and \$M_{t,k}'\$ of \$k\$-subsets' orbits vs. \$t\$-subsets' orbits. For general \$G\$, we show that \$M_{t,k}\$ and \$M_{t,k}'\$ have full rank over \$Q\$, we give a boundon the exponent of the Smith Group of \$M_{t,k}\$ and \$M_{t,k}'\$, and we give a partial answer to the integral preimage problem for \$M_{t,k}\$ and \$M_{t,k}'\$. We propose the Equivariant Sign Conjecture for the matrices \$W_{t,k}(v)\$ using a special basis of the column module of \$W_{t,k}\$ consisting of columns of \$W_{t,k}\$; we verify the Equivariant Sign Conjecture for small cases; and we reduce this conjecture to the case \$v=2k+t\$. For the case \$G=(Z/ nZ)\$, we conjecture that \$M_{t,k}'\$ has a basis of the column module of \$M_{t,k}'\$ that consists of columns of \$M_{t,k}'\$. We prove this conjecture for \$(t,k)=(2,3),(2,4)\$, and we use these results to calculate the Smith Group of \$M_{2,4}\$, \$M_{2,4}'\$,\$M_{2,3}\$, and \$M_{2,3}'\$ for general \$n\$.

Item Type: Thesis (Dissertation (Ph.D.)) Algebraic Codes; Skew Hadamard Difference Sets; Smith Normal Forms California Institute of Technology Physics, Mathematics and Astronomy Mathematics Public (worldwide access) Wilson, Richard M. Wilson, Richard M. (chair)Ku, Cheng YeawWales, David B.Ramakrishnan, Dinakar 5 December 2006 salazc (AT) yahoo.com CaltechETD:etd-05222007-003651 http://resolver.caltech.edu/CaltechETD:etd-05222007-003651 No commercial reproduction, distribution, display or performance rights in this work are provided. 1946 CaltechTHESIS Imported from ETD-db 23 May 2007 26 Dec 2012 02:44

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