John Pugmire, WPI "Blowing up C-Matrices"
Abstract: TBA C-matrices, also known as conference matrices, are matrices satisfying the equation CCT=(n-1)In which are 0 on the main diagonal and +/-1 elsewhere. The existence (or non-existence) of C-matrices has only been shown for certain orders n and is closely related to the Hadamard conjecture. One notable construction is due to Turyn's 1971 paper, On C-Matrices of Arbitrary Powers, which provides a technique for blowing up a C-matrix of order n into a new one of order nm for arbitrary natural numbers m. To this end, Turyn introduces combinatorial objects called G-strings, which have many interesting properties. We present simplified, more explicit proofs of Turyn's results, including new intermediate results on the properties of G-strings.