Linear codes close to the Griesmer bound and the related geometric structures

Abstract

In this paper, we study the behavior of the function $t_q(k)$ defined as the maximal deviation from the Griesmer bound of the optimal length of a linear code with a fixed dimension k:

$$\begin{aligned} t_q(k)=\max _d(n_q(k,d)-g_q(k,d)), \end{aligned}$$

where the maximum is taken over all minimum distances d. Here $n_q(k,d)$ is the shortest length of a q-ary linear code of dimension k and minimum distance d, $g_q(k,d)$ 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 $t_q(k)\rightarrow \infty$ when $k\rightarrow \infty$ which implies that the problem is non-trivial. We prove upper bounds on the function $t_q(k)$. For codes of even dimension k we show that $t_q(k)\le 2(q^{k/2}-1)/(q-1)-(k+q-1)$ which implies that $t_q(k)\in O(q^{k/2})$ for all k. For three-dimensional codes and q even we prove the stronger estimate $t_q(3)\le \log q-1$, as well as $t_q(3)\le \sqrt{q}-1$ for the case q square.