A CHARACTERIZATION OF SOME [N, K, D Q]-CODES MEETING THE GRIESMER BOUND USING A MINIHYPER IN A FINITE PROJECTIVE GEOMETRY

A set F of f points in a finite projective geometry PG(t, q) is called an {f, m; t, q)-minihyper if m (greater-than-or-equal-to 0) is the la rgest integer such that all hyperplanes in PG(t, q) contain at least m points in F. Hamada showed that in the case k greater-than-or-equal-t o 3 and 1 less-than-or-equal-to d < q(k-1), there is a one-to-one corr espondence between the set of all nonequivalent [n, k, d; q]-codes mee ting the Griesmer bound and the set of all {v(k)-n, v(k-1)-n+d; k-1, q }-minihypers where v(l)=(q(l)-1)/(q-1) for any integer l greater-than- or-equal-to 0 (cf. Theorem A.2 in Appendix). This implies that in orde r to characterize all [n,k,d;q]-codes meeting the Griesmer bound for t he case k greater-than-or-equal-to 3 and d=q(k-1)-SIGMA(i=0)k-2epsilon (i)q(i), it is sufficient to characterize all {SIGMA(i=0)k-2epsilon(i) v(i+1); SIGMA(i=0)k-2, k-1, q}-minihypers where 0 less-than-or-equal-t o epsilon(i) less-than-or-equal-to q-1 for i = 0, 1, ..., k-2 (cf. The orem A.3). Recently, many [n, k, d; q]-codes meeting the Griesmer boun d have been characterized by using minihypers. The purpose of this pap er is to provide several fundamental theorems and to survey recent wor k on characterization of minihypers and [n, k, d; q]-codes meeting the Griesmer bound. This is an extended review of Hamada and Deza. With r espect to more recent work, see Hamada (1993).