On difference matrices, transversal designs, resolvable transversal designs and large sets of mutually orthogonal F-squares
For p a prime power and for many values of \(\lambda\), (mixed) difference matrices \(D(k,\lambda;EA(p),Z_ p)\) with \(k=\lambda\) or \(\sim \lambda\) are constructed. These matrices are then used to construct resolvable TD(r,\(\lambda\) ;p) (transversal designs with \(\lambda\) p blocks on r point classes of size p) with \(r=\lambda\) or \(\sim \lambda\). Many of these designs are G-regular. We then use these resolvable TD's to construct many nearly half complete (non-resolvable) TD(r,\(\lambda\) ;p), nearly quarter complete sets of mutually orthogonal F-squares F(\(\lambda\),\(\lambda\) /p), and nearly half complete difference matrices D(k,\(\lambda\) ;EA(p)). In addition, recently constructed generalized Hadamard matrices are used to construct many complete or nearly complete examples of the above combinatorial systems. For example, for all odd prime powers \(p>5\) we show there is a \(TD(r,4p^{m+1};p)\), \(m\geq 0\), with r falling short of the Bose-Bush bound by 1.
scientific article; zbMATH DE number 4077240
scientific article