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

摘要

In this paper, we study the behavior of the function tq(k) defined as the maximal deviation from the Griesmer bound of the optimal length of a linear code with a fixed dimension k: where he maximum is taken over all minimum distances d. Here nq(k,d) is the shortest length of a q-ary linear code of dimension k and minimum distance d, gq(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 tq(k) when k which implies that the problem is non-trivial. We prove upper bounds on the function tq(k). For codes of even dimension k we show that tq(k)2(qk/2-1)/(q-1)-(k+q-1) which implies that tq(k)O(qk/2) for all k. For three-dimensional codes and q even we prove the stronger estimate tq(3)logq-1, as well as tq(3)-1 for the case q square.