On crystal sets

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 nonexistence 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 22n−k. We show that 2− (2n; 4, 1), for n odd, can only exist when 7|n and conjecture that 2-(2n; q2, 1) crystal sets will only exist when q2 + q + 1 is a prime and q2 + q + 1|n.