Symmetric balanced ternary designs with ϱ1 = 1 or 2

Abstract

We prove the following two non-existence theorems for symmetric balanced ternary designs. If ϱ1 = 1 and Λ ≡ 0 (mod 4) then eitherV = Λ + 1 or 4ϱ2 − Λ + 1 is a square and √(4ϱ2 − Λ + 1) divides Λ2 − 1. If ϱ1 = 2 thenV = ((m + 1)/2)2 + 2,K = (m2 + 7)/4 and Λ = ((m − 1)/2)2 + 1 wherem ≡ 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.