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.