A Caltech Library Service

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


Broughton, Wayne Jeremy (1995) Symmetric designs, difference sets, and autocorrelations of finite binary sequences. Dissertation (Ph.D.), California Institute of Technology.


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.

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):
  • Wilson, Richard M.
Thesis Committee:
  • Wilson, Richard M. (chair)
  • Wales, David B.
  • Luxemburg, W. A. J.
  • Doran, William
Defense Date:23 May 1995
Record Number:CaltechETD:etd-09052007-084407
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3338
Deposited By: Imported from ETD-db
Deposited On:11 Sep 2007
Last Modified:26 Dec 2012 02:59

Thesis Files

[img] PDF (Broughton_wj_1995.pdf) - Final Version
Restricted to Caltech community only
See Usage Policy.


Repository Staff Only: item control page