A characterization of some {3v(2)+v(3),3v(1)+v(2);3,3}-minihypers and some [15,4,9;3]-codes with B-2=0

It is known (cf. Hill and Newton (Ars Combin. 25A (1988), 61-72; Designs Codes Cryptography 2 (1992), 137-157) and Remark A.2 in the Appendix) that (1) there is no [14,4,9;3]-code meeting the Griesmer bound and (2) if C is a [15,4,9;3]-code then B-2 = 0 or 2 and (3) there is a one-to-one correspondence between the set of all nonequivalent [15,4,9;3]-codes with B-2 = 0 and the set of all {3v(2) + v(3), 3v(1) + v(2):3,3}-minihypers, where v(1) = 1, v(2) = 4, v(3) = 13 and B-2 denotes the number of codewords of weight 2 in its dual code. Recently it has been shown by Eupen and Hill (Designs Codes Cryptography 4 (1994) 271-282) that a [15,4,9;3]-code with B-2 = 2 is unique up to equivalence. The purpose of this paper is to characterize all [15,4,9;3]-codes with B-2 = 0 using the geometrical structure of the corresponding {3v(2) + v(3), 3v(1) + v(2);3,3]-minihypers. Those results give a complete characterization of [15,4,9;3]-codes.