Geometric invariants for quasi-symmetric designs (Q1113904)

New necessary conditions for the existence of quasi-symmetric 2-designs are derived as follows: Let p be an odd prime and let B be a 2- (v,k,\(\lambda)\) design with block intersection sizes \(s_ 1,s_ 2,...,s_ n\) satisfying \(s_ 1\equiv s_ 2\equiv...\equiv s_ n\equiv s\quad (mod p).\) Then either (1) \(r\equiv \lambda\) (mod \(p^ 2)\), (2) \(v\equiv 0\) (mod 2), \(v\equiv k\equiv s\equiv 0\) (mod p), \((-1)^{v/2}\) is a square in GF(p), (3) \(v\equiv 1\) (mod 2), \(v\equiv k\equiv s\not\equiv 0\) (mod p), \((-1)^{(v-1)/2}s\) is a square in GF(p), (4) \(r\equiv \lambda \equiv 0\) (mod p) and either (a) \(v\equiv 0\) (mod 2), \(v\equiv k\equiv s\not\equiv 0\) (mod p), (b) \(v\equiv 0\) (mod 2), \(k\partial s\not\equiv 0\) (mod p), v/s is a nonsquare in GF(p), (c) \(v\equiv 1\) (mod 2p), \(r\equiv 0\) (mod \(p^ 2)\), \(k\equiv s\not\equiv 0\) (mod p), (d) \(v\equiv p\) (mod 2p), \(k\equiv s\equiv 0\) (mod p), (e) \(v\equiv 1\) (mod 2), \(k\equiv s\equiv 0\) (mod p), v is a nonsquare in GF(p), (f) \(v\equiv 1\) (mod 2), \(k\equiv s\equiv 0\) (mod p), v and \((- 1)^{(v-1)/2}\) are square in GF(p). This shows the non-existence of some quasi-symmetric 2-designs.