Minihypers and linear codes meeting the Griesmer bound: Improvements to results of Hamada, Helleseth and Maekawa
An \(\{f,m;t,q\}\)-minihyper \(F\) is a set of \(f\) points in PG\((t,q)\), where \(t\geq 2\) and \(f\geq 1\), and \(|F\cap H|\geq m\) for every hyperplane \(H\) in PG\((t,q)\) and \(|F\cap H|= m\) for at least one hyperplane \(H\). Minihypers are important in order to characterize codes meeting the Griesmer bound. It is known [\textit{N. Hamada} and \textit{T. Helleseth}, Math. Jap. 38, 925-939 (1993; Zbl 0786.05016); and \textit{N. Hamada} and \textit{T. Maekawa}, Math. Jap. 46, 241-252 (1997; Zbl 0891.94010)] that any \(\{\sum^{t-1}_{i=0}\varepsilon_iv_{i+1},\sum^{t-1}_{i=0}\varepsilon_iv_i;t,q\}\)-minihyper with \(\sum^{t-1}_{i=0}\varepsilon_i = h\) and \((h-1)^2<q\) is the disjoint union of \(\varepsilon_0\) points, \(\varepsilon_1\) lines, \dots, \(\varepsilon_{t-1}(t - 1)\)-dimensional subspaces, where \(v_i = (q^i-1)/(q - 1)\) for any integer \(i\geq 0\). The authors improve the above results by increasing the upper bound on \(h\): (1) for \(q = p^f\), \(p\) prime, \(p > 3\), \(q\) non-square, to \(h \leq q^{6/9}/(1 + q^{1/9})\), (2) for \(q = p^f\) non-square, \(p = 2,3\), to \(h\leq 2^{-1/3}q^{5/9}\), (3) for \(q = p^f\) square, \(p = 2,3\), to \(h\leq \min\{2\sqrt{q} -1,2^{-1/3}q^{5/9}\}\), and (4) for \(q = p^f\) square, \(p\) prime, \(p > 3\), to \(h \leq \min\{2\sqrt q-1, q^{6/9}/(1 + q^{1/9})\}\). When \(q\) is a non-square, the minihyper is the disjoint union of subspaces. When \(q\) is a square, the minihyper is either the disjoint union of subspaces or the disjoint union of subspaces and a subgeometry PG\((l,\sqrt{q})\) of PG\((t,q)\).
scientific article; zbMATH DE number 1746297