A CHARACTERIZATION OF SOME (V2+2V3, V1+2V2-K-1,3)-MINIHYPERS AND SOME (VK-30, K, 3K-1-21-3)-CODES MEETING THE GRIESMER BOUND

Let F be a set of f points in a finite projective geometry PG(t,q) of t dimensions where t greater-than-or-equal-to 2, f greater-than-or-equal-to 1 and q is a prime power. If (a) \F and H\greater-than-or-equal-to m for any hyperplane H in PG(t,q) and (b) \F and H\=m for some hyperplane H in PG(t,q), then F is called an {fm;t,q}-minihyper where m greater-than-or-equal-to 0 and \A\ denotes the number of elements in the set A. Tamari (1981, 1984) characterized all {v(alpha+1), v(alpha);t,q}-minihypers where 0<alpha<t and v(l)=(q(l)-1)/(q-1) for any integer l greater-than-or-equal-to 0. Recently, Hamada and Deza (1988b) and Hamada and Helleseth (1990a) characterized all {v(alpha+1)+v(beta+1)+v(gamma+1),v(alpha)+v(beta)+v(gamma);t,q}-minihypers for any integers t, q, alpha, beta and gamma such that q greater-than-or-equal-to 5 and 0 less-than-or-equal-to alpha less-than-or-equal-to beta less-than-or-equal-to gamma < t. The purpose of this paper is to characterize all {v(alpha+1)+v(beta+1)+v(gamma+1),v(alpha)+v(beta)+v(gamma);t,q}-minihypers for the case t greater-than-or-equal-to 3, alpha=1, beta=gamma=2 and q=3. Using those results, (1) a geometrical characterization of 10-caps in PG(3,3) is given and (2) all (n,k,d,3)-codes meeting the Griesmer bound are characterized for the case k greater-than-or-equal-to 4 and d=3k-1-21.