A class of non-embeddable designs
The authors show that a necessary condition for a 2-\((v,k,k-1)\)-design which contains a 2-\((v_ 0,k,k-1)\)-subdesign to be embeddable as a derived design into a symmetric 2-\((1+v(v-1)/k,v,k)\)-design is the following inequality: \[ v_ 0(v_ 0-1)((v_ 0-1)(v_ 0-k)-2(v- k))+2(v-1)(v-k)\geq 0. \] They use this criterion to construct three classes of non-embeddable quasi-derived designs. E.g., if \(n^ 2-n-1\) is a prime power and \(n>2\), then there exists a non-embeddable 2-\((n^ 2,n,1)\)-design. As an application, this also yields the existence of quasi-multiples of affine planes of non-prime power order \(n\) with small \(\lambda\)-value, improving results given by the reviewer in his paper [On the existence of small quasimultiples of affine and projective planes of arbitrary order, Discrete Math. 85, No. 2, 177-189 (1990; Zbl 0729.05009)].
scientific article; zbMATH DE number 221987
scientific article