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

Citation

Broughton, Wayne Jeremy (1995) Symmetric designs, difference sets, and autocorrelations of finite binary sequences. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/m1he-d221. https://resolver.caltech.edu/CaltechETD:etd-09052007-084407

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 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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