Additive symmetric bijections on geometric structures (Q1066905)

For a geometric structure \((\Omega,B)\) a bijection \(\phi: \Omega\to B\) is said to be symmetric when \(x\in\phi(y)\) iff \(y\in\phi(x)\), and if \(\Omega\) has a vector space structure and \(B\) is a collection of subspaces, then \(\phi\) is additive if \(\phi(x)\cap\phi(y)\subseteq \phi(x+y)\). Additive symmetric bijections arise naturally in quadratic form theory where \(\Omega\) is the group of square classes of a field viewed as \(\mathrm{GF}(2)\)-vector space and \(B\) is the collection of value groups \(B_ x\) of binary forms \(\langle 1,-x\rangle\). Then in most cases \(\phi(x)=B_ x\) is a symmetric additive bijection. The author introduces a class of block designs called IP-designs, satisfying certain intersection property, and gives a construction of additive symmetric bijections on IP-designs. He also proves that additive symmetric bijections exist on a design iff the design arises as points and hyperplanes of a projective geometry over \(\mathrm{GF}(2)\).