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:etd-09052007-084407
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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 semi-primitive, 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|
|Thesis Availability:||Restricted to Caltech community only|
|Defense Date:||23 May 1995|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||11 Sep 2007|
|Last Modified:||26 Dec 2012 02:59|
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