On the Sharpness of the Griesmer Bound

We investigate the following version of the main problem of coding theory: Given the integer k and the prime power q, what is the value oftq(k):=maxd⁡nq(k,d)−gq(k,d). We give several formulations of this problem: in terms of linear codes, arcs and minihypers. We provide general constructions that give upper bounds on tq(k).