On a conjecture of jungnickel and tonchev for quasi-symmetric designs

Jungnickel and Tonchev conjectured in 4] that if a quasi-symmetric design D is an s-fold quasi-multiple of a symmetric (v,k,λ) design with (k,(s ? 1)λ) = 1, then D is a multiple. We prove this conjecture under any one of the conditions: s ≤ 7, k ? 1 is prime, or the design D is a 3-design. It is shown that for any fixed s, the conjecture is true with at most finitely many exceptions. The unique quasi-symmetric 3-(22,7,4) design is characterized as the only quasi-symmetric 3-design, which as a 2-design is an s-fold quasi-multiple with s ≡ 1 (mod k). © 1994 John Wiley & Sons, Inc.