A tour through design theory

Abstract

In this talk, we will visit three types of designs (sequences with perfect autocorrelation, relative difference sets, and Hadamard matrices). The periodic autocorrelation of a sequence is a measure for how much the sequence differs from its cyclic shifts. If the autocorrelation values for all nontrivial cyclic shifts are 0, then the sequence is perfect. It is very difficult to construct perfect sequences over 2^nd, 4^th, and in general over $n$^th roots of unity. It is conjectured that perfect sequences over n-th roots of unity do not exist for lengths greater than $n^2$. Due to the difficulty to construct perfect sequences over n-th roots of unity, there has been some focus on other classes of sequences with good autocorrelation. One of these classes is the family of perfect sequences over the quaternions. In this talk, I will introduce perfect sequences over the quaternion groups $Q_8$ and $Q_{24}$. Such sequences exhibit interesting symmetry patterns, which I aim to explain via a connection with relative difference sets and Hadamard matrices.