Characterization of \(\{2(q+1) + 2, 2; t, q\}\)-minihypers in \(PG(t,q) (t\geqslant 3, q\in \{3,4\})\)
An \(\{f,m;t,q\}\)-minihyper is a set \(F\) of \(f\) points in \(PG(t,q)\) such that \(F\) meets every hyperplane in at least \(m\) points and at least one hyperplane in exactly \(m\) points. Minihypers have connections with codes and the problem of characterizing all \([n,k,d,q]\)-codes (such a code is a \(k\)-dimensional subspace of an \(n\)-dimensional vector space over \(GF(q)\) such that every nonzero vector has Hamming weight at least \(d)\) meeting the Griesmer bound, i.e. such that \(n=\sum^{k-1}_{i=0}\left[{d\over q^ i}\right]\), lead the authors to the problem of characterizing the minihypers in the title. Under other restrictions, a characterization exists already (e.g. \(t\geq 3\) and \(q\geq 5\) was done by the first author and \textit{M. Deza} [Discrete Math. 71, No. 3, 219-231 (1988; Zbl 0678.94010)]). The main result of the paper under review is the fact that there exist no irreducible \(\{12,2;t,4\}\)-minihypers and that there exists exactly one class of irreducible \(\{10,2;t,3\}\)-minihypers. Irreducible means that the minihyper is not a union of at least two disjoint minihypers.
scientific article; zbMATH DE number 205599