Symmetric designs, difference sets, and autocorrelations of finite binary sequences

Author: Broughton, Wayne Jeremy

Year: 1995

Degree: Dissertation (Ph.D.)

Advisor: Wilson, Richard M.

Committee Members: Wilson, Richard M.; Wales, David B.; Luxemburg, W. A. J.; Doran, William

Option: Mathematics

DOI: 10.7907/m1he-d221

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

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.

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