xistence of \(V_\lambda(m,t)\) vectors

Colbourn introduced \(V_\lambda(m, t)\) to construct transversal designs with index \(\lambda\). A \(V_\lambda(m, t)\) leads to a \((mt + 1, mt + 2; \lambda,0; t)\)-aussie-difference matrix. In this article, we use Weil’s theorem on character sums to show that for any integer \(\lambda \geq 2\), a \(V_\lambda(m, t)\) always exists in \(GF(mt + 1)\) for any prime power \(mt+1 > B_\lambda(m) = \left[\frac{E+\sqrt{E^2+4F}}{2}\right]^2\), where \(E = \lambda(u-1)(m-1)m^u-m^{u-1}+1,F=(u-1)\lambda m^u\) and \(u = \left\lfloor\frac{m{\lambda}+1+(-1)^{\lambda+1}}{2}\right\rfloor\). In particular, we determine the existence of \(V_{\lambda}(m, t)\) for \((\lambda, m) = (2, 2), (2, 3)\).