In this paper, we study the behavior of the function tq(k)\documentclass[12pt]{minimal}
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\begin{document}$$t_q(k)$$\end{document} defined as the maximal deviation from the Griesmer bound of the optimal length of a linear code with a fixed dimension k: tq(k)=maxd(nq(k,d)-gq(k,d)),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} t_q(k)=\max _d(n_q(k,d)-g_q(k,d)), \end{aligned}$$\end{document}where the maximum is taken over all minimum distances d. Here nq(k,d)\documentclass[12pt]{minimal}
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\begin{document}$$n_q(k,d)$$\end{document} is the shortest length of a q-ary linear code of dimension k and minimum distance d, gq(k,d)\documentclass[12pt]{minimal}
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\begin{document}$$g_q(k,d)$$\end{document} is the Griesmer bound for a code of dimension k and minimum distance d. We give two equivalent geometric versions of this problem in terms of arcs and minihypers. We prove that tq(k)→∞\documentclass[12pt]{minimal}
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\begin{document}$$t_q(k)\rightarrow \infty $$\end{document} when k→∞\documentclass[12pt]{minimal}
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\begin{document}$$k\rightarrow \infty $$\end{document} which implies that the problem is non-trivial. We prove upper bounds on the function tq(k)\documentclass[12pt]{minimal}
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\begin{document}$$t_q(k)$$\end{document}. For codes of even dimension k we show that tq(k)≤2(qk/2-1)/(q-1)-(k+q-1)\documentclass[12pt]{minimal}
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\begin{document}$$t_q(k)\le 2(q^{k/2}-1)/(q-1)-(k+q-1)$$\end{document} which implies that tq(k)∈O(qk/2)\documentclass[12pt]{minimal}
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\begin{document}$$t_q(k)\in O(q^{k/2})$$\end{document} for all k. For three-dimensional codes and q even we prove the stronger estimate tq(3)≤logq-1\documentclass[12pt]{minimal}
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\begin{document}$$t_q(3)\le \log q-1$$\end{document}, as well as tq(3)≤q-1\documentclass[12pt]{minimal}
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\begin{document}$$t_q(3)\le \sqrt{q}-1$$\end{document} for the case q square.
