A series of Siamese twin designs (Q998529)

A symmetric \((v,k,\lambda)\) design is a finite incidence structure \(({\mathcal P},{\mathcal B},{\mathcal I})\) where \(\mathcal P\) and \(\mathcal B\) are disjoint sets and \({\mathcal I} \subseteq {\mathcal P} \times {\mathcal B}\), with the following properties: (1) \(|{\mathcal P}| = |{\mathcal B}| =v\); (2) every element of \(\mathcal B\) is incident with exactly \(k\) elements of \(\mathcal P\); (3) every pair of distinct elements of \(\mathcal P\) is incident with exactly \(\lambda\) elements of \(\mathcal B\). The elements of the set \(\mathcal P\) are called points and the elements of the set \(\mathcal B\) are called blocks. A \(\{0,\pm 1\}\)-matrix \(S\) is called a Siamese twin design sharing the entries of \(I\) if \(S=I+K - L\), where \(I,K,L\) are nonzero \(\{0,1\}\)-matrices and both \(I+K\) and \(I+L\) are incidence matrices of symmetric designs with the same parameters. In this note, the author constructs Siamese twin design swith parameters \((4(p+1)^{2},2p^{2}+3p+1,p^{2}+p)\) where \(p\) and \(2p+3\) are prime powers and \(p\equiv 3\mod 4\).