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# Brock TR # CS-03-10 Abstract

*Investigation of Perfect Ternary Arrays PTA(60,25) * [PDF]

P.Becker, S.Houghten and W.Haas, October 2003.

Perfect ternary arrays are closely related to difference sets and group invariant weighing matrices. A previous paper by one of the authors demonstrated strong restrictions on difference sets with parameters (120, 35, 10). A perfect ternary array with energy 25 and order 60, PTA(60,25), would obey an equation analogous to the difference set equation for (120, 35, 10). A perfect ternary array PTA(n,k) is equivalent to a group-developed weighing matrix in a group of order n. There is one known example of a weighing matrix developed over a nonsolvable group of order 60; no solvable examples are known. In this paper, we describe a search for weighing matrices (and corresponding perfect ternary arrays) developed over solvable groups of order 60. We analyze the quotient structure for each group. Techniques from representation theory, including a new viewpoint on complementary quotient images, are used to restrict possible weighing matrices. Finally, we describe a partially completed computer search for such matrices and arrays.