An infinite family of non-embeddable quasi-residual designs with \(k<v/2\)
Until not too long ago, most non-embeddable quasi-residual designs had the parameters of the Bhattacharya design or their non-embeddability was based on the non-existence of the corresponding symmetric design. The first constructions of other examples had large blocksize. In the past decade, examples have been constructed with blocksize smaller than half the number of points. In this paper an infinite family of such designs is constructed. The parameters are 2-\((2(3^{d+ 1})- 2, 2(3^d), 3^d)\), where \(d\geq 1\). By a glueing construction, using blocks of the 2-\((q^{d+ 1}- q+ 1, q^d, q^{d- 1})\) design constructed by \textit{C. J. Mitchell} [An infinite family of symmetric designs, Discrete Math. 26, 247-240 (1979; Zbl 0416.05016)] and \(q\) copies of \(\text{AG}_{d- 1}(d, q)\), a design with parameters \(2q^{d+ 1}- q+ 1\), \(2q^d\), \(q^d\) is constructed. The case \(q= 3\) yields the non-embeddable family. The reading is slightly complicated by the fact that the author refers to an (as yet) unpublished paper of hers (manuscript available).
scientific article; zbMATH DE number 883821