Nonexistence of certain pseudogeometric graphs

Izo(r) is the pseudogeometric graph for pGr(s,t) which satisfies the Krein equality and of which the local graph also satisfies the Krein equality. We prove that Izo(r) exists if and only if a spherical tight 4-design on S(2r+1)2−4 exists. Nonexistence of infinitely many spherical tight 4-designs is well known. It gives the nonexistence of Izo(r) for infinitely many r. Especially, Izo(4) does not exist, which was the first open case.