A set F of f points in a finite projective geometry PG(t,q) is called an {f,m;t,q}-minihyper if m (⩾0) is the largest integer such that all hyperplanes in PG(t,q) contain at least m points in F. Hamada showed that in the case k⩾3 and 1⩽d<qk−1, there is a one-to-one correspondence between the set of all nonequivalent [n,k,d;q]-codes meeting the Griesmer bound and the set of all {vk−n,vk−1−n+d;k−1,q}-minihypers where for any integer l⩾0 (cf. Theorem A.2 in Appendix). This implies that in order to characterize all [n,k,d;q]-codes meeting the Griesmer bound for the case k⩾3 and d = qk-1−∑k-2i=0εiqi, it is sufficient to characterize all {∑k−2i=0εivi+1,∑k−2i=0εivi;k − 1,q} -minihypers where 0⩽εi⩽q − 1 for i = 0, 1,…,k − 2 (cf. Theorem A.3).
Recently, many [n,k,d;q]-codes meeting the Griesmer bound have been characterized by using minihypers. The purpose of this paper is to provide several fundamental theorems and to survey recent work on characterization of minihypers and [n,k,d;q]-codes meeting the Griesmer bound. This is an extended review of Hamada and Deza. With respect to more recent work, see Hamada (1993).
