On crystal sets

Authors

Chun Li, University of Wollongong
Jennifer Seberry, University of WollongongFollow

RIS ID

52055

Publication Details

Li, C. & Seberry, J. (2012). On crystal sets. Australasian Journal of Combinatorics, 52 89-110.

Abstract

We are interested in 2-crystal sets and protocrystal sets in which every difference between distinct elements occurs zero or an even number of times. We show that several infinite families of such sets exist. We also give non-existence theorems for infinite families. We find conditions to limit the computer search space for such sets. We note that search for 2-crystal sets (n; k₁, k₂), k =k₁ + k₂ even, in a set of size n, immediately cuts the search space for two circulant weighing matrices with periodic autocorrelation function zero from 3²ⁿ to 2^2n-k. We show that 2-(2n; 4,1), for n odd, can only exist when 71n and conjecture that 2-(2n; q², 1) crystal sets will only exist when q² + q + 1 is a prime and (q² + q + 1) In.