Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

Abstract

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any $\left\{ {\sum\nolimits_{i = 0}^{t - 1} {{\varepsilon }_i {\upsilon }_{i + 1} ,\sum\nolimits_{i = 0}^{t - 1} {{\varepsilon }_i {\upsilon }_i ;t,q} } } \right\}$-minihyper, with $\sum\nolimits_{i = 0}^{t - 1} {{\varepsilon }_i = h}$, where $\left( {h - 1} \right)^2 < q$, is the disjoint union of ${{\varepsilon }_0 }$ points, ${{\varepsilon }_1 }$ lines,..., ${\varepsilon }_{t - 1} \left( {t - 1} \right)$-dimensional subspaces. For q large, we improve on this result by increasing the upper bound on $h:\left( 1 \right){\text{ for }}q = p^f ,p{\text{ prime, }}p > 3,{\text{ }}q$ non-square, to $h \leqslant q^{6/9} /\left( {1 + q^{1/9} } \right),\left( 2 \right){\text{ for }}q = p^f$ non-square, $p = 2,3,{\text{ to }}h \leqslant 2^{ - 1/3} q^{5/9} ,\left( 3 \right){\text{ for }}q = p^f$ square, $p = 2,3,{\text{ to }}h \leqslant {\text{ min }}\left\{ {2\sqrt q - 1,2^{ - 1/3} q^{5/9} } \right\}$, and (4) for $q = p^f$ square, p prime, p<3, to $h \leqslant {\text{ min }}\left\{ {2\sqrt q - 1,q^{6/9} /\left( {1 + q^{1/9} } \right)} \right\}$. In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry $PG\left( {l,\sqrt q } \right){\text{ of }}PG\left( {t,q} \right)$. For the coding-theoretical problem, our results classify the corresponding $\left[ {n = {\upsilon }_{t + 1} - \sum\nolimits_{i = 0}^{t - 1} {{\varepsilon }_i {\upsilon }_{i + 1} ,k = t + 1,d = q^t - } \sum\nolimits_{i = 0}^{t - 1} {q^i {\varepsilon }_i ;q} } \right]$ codes meeting the Griesmer bound.