Weighted {δ(q+1),δ;k-1,q}{δ(q+1),δ;k-1,q}-minihypers

In Ferret and Storme [A classification result on weighted {δ(p3+1),δ;3,p3}{δ(p3+1),δ;3,p3}-minihypers, J. Combin. Designs 12 (2004) 197–220; A classification result on weighted {δvμ+1,δvμ;N,p3}{δvμ+1,δvμ;N,p3}-minihypers, Discrete Appl. Math. 154 (2004) 277–293], Govaerts and Storme [On a particular class of minihypers and its applications. II. Improvements for q square, J. Combin. Theory Ser. A 97 (2) (2002) 369–393; On a particular class of minihypers and its applications. I. The result for general q  , Designs Codes Cryptogr. 28 (2003) 51–63] and Govaerts, Storme and Van Maldeghem [On a particular class of minihypers and its applications. III. Applications, European J. Combin. 23 (2002) 659–672], weighted {δvμ+1,δvμ;N,q}{δvμ+1,δvμ;N,q}-minihypers were classified. This class of minihypers is, next to being interesting for classifying linear codes meeting the Griesmer bound, a very important geometrical structure for solving problems in finite projective spaces. In Ferret and Storme [A classification result on weighted {δ(p3+1),δ;3,p3}{δ(p3+1),δ;3,p3}-minihypers, J. Combin. Designs 12 (2004) 197–220; A classification result on weighted {δvμ+1,δvμ;N,p3}{δvμ+1,δvμ;N,p3}-minihypers, Discrete Appl. Math. 154 (2004) 277–293], Govaerts and Storme [On a particular class of minihypers and its applications. II. Improvements for q square, J. Combin. Theory Ser. A 97 (2) (2002) 369–393] and Govaerts, Storme and Van Maldeghem [On a particular class of minihypers and its applications. III. Applications, European J. Combin. 23 (2002) 659–672], there were restrictions on the weights of the points of the minihypers; in Govaerts and Storme [On a particular class of minihypers and its applications. I. The result for general q  , Designs Codes Cryptogr. 28 (2003) 51–63], there were no restrictions on the weights of the points, but the results were only valid for δ⩽εδ⩽ε, with q+1+εq+1+ε the size of the smallest non-trivial blocking sets in PG(2,q)PG(2,q). In this article, we improve this latter result for weighted {δ(q+1),δ;N,q}{δ(q+1),δ;N,q}-minihypers, without restrictions on the weights of the points. The largest improvements are obtained for q=p2q=p2, p   prime, p⩾11p⩾11, where we increase the upper bound to δ⩽(q-1)/4δ⩽(q-1)/4.