Spectrum of splitting 3-designs with block size \(3 \times 2\) (Q2831590)

Splitting \(t\)-designs arise in codes for authentication. A \(t\)-\((v,u \times k, \lambda)\) splitting design is a set \(X\) of \(v\) points and a collection \({\mathcal B}\) of subsets of \(X\) (blocks), each of size \(uk\). Each block \(B\) is equipped with a partition into \(u\) subblocks each of size \(k\). The defining property is that the points of each \(t\)-subset of \(X\) appears in exactly \(\lambda\) blocks in such a way that no two elements of the \(t\)-subset appear in the same subblock. In this paper, a complete existence result for \(3\)-\((v,3 \times 2,\lambda)\) splitting designs is established.