Recursive constructions of arrays with low autocorrelation

2017-03-01T01:34:40Z (GMT) by Jolly, Nathan James
Matrices with low autocorrelation are very useful in applications such as digital watermarking. This thesis introduces a new class of matrices with low autocorrelation, and presents a new method of constructing such matrices to unbounded sizes. In 2001, Arasu and de Launey presented several recursive constructions of perfect and what they call "almost perfect" arrays over the quaternary (4th roots of unity) alphabet. Recursive means that under certain conditions, the construction can be applied repeatedly, thereby generating larger and larger arrays while preserving low autocorrelation. Arasu and de Launey employed Laurent polynomials over two indeterminates to state and prove the recursive constructions. While this policy made the proofs expedient, it also obfuscated the simple yet powerful nature of the constructions themselves. In this thesis, we present some of those recursive constructions in the language of arrays, with no recourse to polynomials. This research direction has led to three new contributions: (1) A translation of an algorithm by Arasu and de Launey to "inflate" perfect quaternary arrays, leading to joint work with Barerra Acevedo giving infinitely large perfect arrays over the basic quaternions; (2) A new algebra of arrays involving array convolution as the multiplicative operation as well as special functions called "index functions", culminating in a simpler statement and proof of Arasu and de Launey's recursive construction of almost perfect arrays; (3) A new recursive construction of what we call "n-perfect" arrays, which extends both the concept and recursive construction of Arasu and de Launey's almost perfect arrays.