Investigation of Perfect Ternary Arrays PTA ( 60 , 25 )

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. 1 Definitions and examples There are a variety of combinatorial objects which satisfy the general matrix equation: MM = kI. Some of these objects can be further described as group-developed; that is, their internal structure is most easily reproduced via multiplication in a finite group. In this paper, we describe a search for a particular instance of such objects, weighing matrices with order 60 and weight 25. Such matrices are a generalization of perfect ternary arrays with order 60 and energy 25. We also discuss the closely related concepts of difference sets, obtaining further restrictions on (120, 35, 10) difference sets. Definition 1 (Perfect Ternary Array – PTA). An r-dimensional ternary array is a s1 × s2 × ... × sr array with entries chosen from the set {0, 1,−1}. Such an array is perfect if its out-of-phase periodic autocorrelation coefficients are zero. Definition 2 (Parameters of Perfect Ternary Arrays). Assume A is a s1×s2×· · ·×sr PTA. The energy k = e(A) of A is the number of nonzero entries in A. The order of A is the product n = Πsi. The energy efficiency of A is the ratio k/n. A is described as a PTA(n,k). Perfect ternary arrays are of particular interest in communications theory. The autocorrelation properties of these arrays are their salient feature. For our work, however, it is easier to deal with equivalent matrix structures; the autocorrelation properties translate to orthogonality of matrix rows. Definition 3 (Group Developed Weighing Matrix). A weighing matrix W(n,k) of order n with weight k is an n×n matrix with entries from {0, 1,−1} such that WW = kIn. Such a matrix is group-developed under G if the rows and columns can be indexed by elements of G so that wg,h = wgf,hf for all g, h, f in G. A variety of constructions are known for weighing matrices; the best summary can be found in [6]. These constructions, however, rarely result in group-developed matrices – the natural generalization of perfect ternary arrays. Proposition 1 (Arasu). Assume A is a s1 × · · · × sr PTA(n,k). Let G be the group Z1 × Z2 × · · · × Zr. A is equivalent to a G-developed weighing matrix W (|G|, k). It is also equivalent to a disjoint pair of sets P, N in G such that (P −N)(P −N)(−1) = k.