Quasi-symmetric 2,3,4-designs
Quasi-symmetric designs are block designs with two block intersection numbers x and y. In this paper, it is shown that with the exception of \((x,y)=(0,1)\), for a fixed value of the block size k, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positive x and y, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions of v, k and \(\lambda\). Using this quadratic, various characterizations of the Witt-Lüneburg design on 23 points are obtained. It is also shown that if \(x=1\), then a fixed value of \(\lambda\) determines at most finitely many such designs. One of the interesting results is as follows: Let D be a quasi-symmetric 3-(v,k,\(\lambda)\) design with \(x=1\). Then (i) v-2 divides \(k(k-1)^ 2(k-2)\); (ii) the larger intersection number \(y\geq 3\) with equality if and only if D is the unique Witt-Lüneburg 4-(23,7,1) design or its residual a 3-(22,7,4) design; (iii) \(\lambda\geq y\); (iv) \(\lambda_ 3\geq 4\) with equality if and only if D is the 3-(22,7,4) design; (v) \(\lambda_ 3=5\) if and only if D is the Witt-Lüneburg design on 23 points; (vi) \(\lambda_ 3\leq k- 2\) with equality if and only if D is the Witt-Lüneburg design on 23 points; (vii) \(\lambda_ 2\leq v-2\) with equality if and only if D is the Witt-Lüneburg design on 23 points; (viii) v is bounded below and above as follows, \[ \frac{(k-2)^ 2(y-1)}{y(y-2)}\quad +\quad \frac{k- 2}{y}\quad -\quad 1\quad \leq \quad v-3\quad \leq \quad \frac{(k-2)(k- 3)}{y-2}, \] further, in which the upper bound is sharp with equality if and only if D is the Witt-Lüneburg 4-design on 23 points. Two conjectures on the classification of quasi-symmetric 3-designs are interestingly stated.
scientific article; zbMATH DE number 4068885