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 \documentclass[12pt]{minimal}
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$$\left\{ {\sum\nolimits_{i = 0}^{t - 1} {{\varepsilon }_i {\upsilon }_{i + 1} ,\sum\nolimits_{i = 0}^{t - 1} {{\varepsilon }_i {\upsilon }_i ;t,q} } } \right\}$$
\end{document}-minihyper, with \documentclass[12pt]{minimal}
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$$\sum\nolimits_{i = 0}^{t - 1} {{\varepsilon }_i = h}$$
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$$\left( {h - 1} \right)^2 < q$$
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$${{\varepsilon }_0 }$$
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$${{\varepsilon }_1 }$$
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$${\varepsilon }_{t - 1} \left( {t - 1} \right)$$
\end{document}-dimensional subspaces. For q large, we improve on this result by increasing the upper bound on \documentclass[12pt]{minimal}
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$$h:\left( 1 \right){\text{ for }}q = p^f ,p{\text{ prime, }}p > 3,{\text{ }}q$$
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$$h \leqslant q^{6/9} /\left( {1 + q^{1/9} } \right),\left( 2 \right){\text{ for }}q = p^f$$
\end{document} non-square, \documentclass[12pt]{minimal}
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$$p = 2,3,{\text{ to }}h \leqslant 2^{ - 1/3} q^{5/9} ,\left( 3 \right){\text{ for }}q = p^f$$
\end{document} square, \documentclass[12pt]{minimal}
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$$p = 2,3,{\text{ to }}h \leqslant {\text{ min }}\left\{ {2\sqrt q - 1,2^{ - 1/3} q^{5/9} } \right\}$$
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$$q = p^f$$
\end{document} square, p prime, p<3, to \documentclass[12pt]{minimal}
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$$h \leqslant {\text{ min }}\left\{ {2\sqrt q - 1,q^{6/9} /\left( {1 + q^{1/9} } \right)} \right\}$$
\end{document}. 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 \documentclass[12pt]{minimal}
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$$PG\left( {l,\sqrt q } \right){\text{ of }}PG\left( {t,q} \right)$$
\end{document}. For the coding-theoretical problem, our results classify the corresponding \documentclass[12pt]{minimal}
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$$\left[ {n = {\upsilon }_{t + 1} - \sum\nolimits_{i = 0}^{t - 1} {{\varepsilon }_i {\upsilon }_{i + 1} ,k = t + 1,d = q^t - } \sum\nolimits_{i = 0}^{t - 1} {q^i {\varepsilon }_i ;q} } \right]$$
\end{document} codes meeting the Griesmer bound.
