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A weighing matrix \( W(n, k) \) of order \( n \) with weight \( k \) is an \( n \times n \) matrix with entries from \( \{0, 1, -1\} \) which satisfies \( WW^T = kI_n \). Such a matrix is group-developed if its rows and columns can be indexed by elements of a finite group \( G \) so that \( w_{g,h} = w_{gf,hf} \) for all \( g,h,f \) in \( G \). Group-developed weighing matrices are a natural generalization of perfect ternary arrays and Hadamard matrices. They are closely related to difference sets.
We describe a search for weighing matrices with order 60 and weight 25, developed over solvable groups. There is one known example of a \( W(60, 25) \) developed over a non-solvable group; no solvable examples are known.
We use techniques from representation theory, including a new viewpoint on complementary quotient images, to restrict solvable examples. We describe a computer search strategy which has eliminated two of twelve possible cases. We summarize plans to complete the search.