Autocorrelation, difference sets and
the equation XX^T=36

A variety of digital technologies (radar, wireless communication, etc.) require the construction of signals with special autocorrelation properties. Of special interest are sequences X with periodic autocorrelation d² (so that XX* = d²; the sum of the entries of X is d.) Since the autocorrelation is periodic, we may view X as an element of the group ring Z[G], where G is cyclic, and use the rational idempotents of G to construct X.
More generally, the group ring equation XX^T= d²; XG=dG (where G is any finite group) appears in a variety of problems in algebraic combinatorics, including multiplier theorems of difference sets (where the equation is called the "CH-equation.") The equation also appears in an analysis of homomorphic images of difference sets and relative difference sets.
The CH-equation with d=6 in groups G of order 44 arose in two recent difference sets problems: a search for (176, 50, 14) difference sets (initiated by undergraduate student, Oliver Gjoneski, summer NSF-REU 2004) and a search for (704, 38, 2) difference sets (recent work by Solomon Osifodunrin, Central Michigan University.) Using algebraic number theory, group representations and rational idempotents of the group ring, we find all solutions to this equation in certain groups. Along the way, we discuss the use of rational idempotents in combinatorial constructions in various finite groups.