On a particular class of minihypers and its applications. I: The result for general \(q\)
A partial \(s\)-spread \(S\) of \(\Sigma = \text{PG}(d,q)\) is a collection of mutually skew \(s\)-dimensional subspaces in \(\Sigma\). If all points of \(\Sigma\) are covered, \(S\) is called an \(s\)-spread; this is possible if and only if \((s + 1)|(d + 1)\). A partial spread which is not a spread is said to have positive deficiency. A blocking set \(B\) in \(\pi = \text{PG}(2,q)\) is a set of points in \(\pi\) such that every line of \(\pi\) meets \(B\) in at least one point. A blocking set is called nontrivial if it does not contain a line. Upper bounds on the size of maximal partial 1-spreads of \(\text{PG}(3,q)\) with positive deficiency have recently been improved by using an important link with blocking sets of \(\text{PG}(2,q)\). In the paper under review similar improvements are made for maximal partial \(s\)-spreads of positive deficiency whenever \((s+1)|(d+1)\), again by exploiting a link with nontrivial blocking sets of \(\text{PG}(2,q)\). The key ingredient is the use of (weighted) minihypers, first introduced in [\textit{N. Hamada} and \textit{T. Helleseth}, Math. Jap. 38, 925-939 (1993; Zbl 0786.05016)]. The results are then generalized to \(s\)-covers of \(\text{PG}(d,q)\); that is, subsets of s-dimensional subspaces of \(\text{PG}(d,q)\) which cover every point.
scientific article; zbMATH DE number 1892218