Citation
Broughton, Wayne Jeremy (1995) Symmetric designs, difference sets, and autocorrelations of finite binary sequences. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd09052007084407
Abstract
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Symmetric designs with parameters [...] are very regular structures useful in the design of experiments, and [...]difference sets are a common means of constructing them, as well as being interesting subsets of groups in their own right. We investigate [...]symmetric designs and [...]difference sets, especially those satisfying v = 4n + 1, where n = k [...]. These must have parameters of the form [...], [...], [...] for [...]. Such difference sets exist for t = 1, 2. Generalizing work of M. Kervaire and S. Eliahou, we conjecture that abelian difference sets with these parameters do not exist for [...], and we prove this for large families of values of t (or n). In particular, we eliminate all values of t except for a set of density 0. The theory of biquadratic reciprocity is especially useful here, to determine whether or not certain primes are biquadratic, and hence semiprimitive, modulo the factors of v.
Cyclic difference sets also correspond to [...]sequences of length v with constant periodic autocorrelation. Such sequences are of interest in communications theory, especially if they also have small aperiodic correlations. We find and prove bounds on the aperiodic correlations of a binary sequence that has constant periodic correlations. As an example, such sequences corresponding to cyclic difference sets satisfying v = 4n have aperiodic autocorrelations bounded absolutely by (3/2)n, with a similar bound in the case of cyclic Hadamard difference sets (those satisfying v = 4n  1).
Finally, we present an alternative construction of a class of symmetric designs due to A.E. Brouwer which includes the v = 4n + 1 designs as a special case.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Restricted to Caltech community only 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  23 May 1995 
Record Number:  CaltechETD:etd09052007084407 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd09052007084407 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3338 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  11 Sep 2007 
Last Modified:  26 Dec 2012 02:59 
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