Citation
SalazarLazaro, Carlos Harold (2007) Association Schemes, Codes, and Difference Sets. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8AMDKY33. https://resolver.caltech.edu/CaltechETD:etd05222007003651
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 nadic codes; generalized skew hadamard difference sets; and equivariant incidence structures.
In the second chapter, we introduce nadic codes, a generalization of the Duadic Codes studied by Pless and Rushanan, and we solve the corresponding existence problem. We introduce nadic groups, canonical pplitters, and Margarita Codes to generalize the "selfdual" 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,G1}, where G_{1}=(Z/exp(G)Z)* and G is a group, to reduce the existence problem to an integral equation. Using a special finitedimensional algebra or Association Scheme, we show A_{G,G1}^{2} = G/pI for general G. With the aid of A_{G,G1}, we show some necessary conditions for the existence of a GSHDS D in the group (Z/pZ) x (Z/p^{2}Z)^{2α+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 tsubsets of {1,...,v} vs. ksubsets of {1,...,v}. Also, given a group G acting on {1,...,v}, we define analogous incidence matrices M_{t,k} and M'_{t,k} of ksubsets' orbits vs. tsubsets' orbits. For general G, we show that M_{t,k} and M'_{t,k} have full rank over Q, we give a bound on 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.))  

Subject Keywords:  Algebraic Codes; Skew Hadamard Difference Sets; Smith Normal Forms  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  5 December 2006  
NonCaltech Author Email:  salazarlazc (AT) gmail.com  
Record Number:  CaltechETD:etd05222007003651  
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd05222007003651  
DOI:  10.7907/8AMDKY33  
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  1946  
Collection:  CaltechTHESIS  
Deposited By:  Imported from ETDdb  
Deposited On:  23 May 2007  
Last Modified:  26 Feb 2020 22:35 
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