Weighted image-minihypers Original Research Article

In Ferret and Storme [A classification result on weighted image-minihypers, J. Combin. Designs 12 (2004) 197–220; A classification result on weighted image-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 image-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 image-minihypers, J. Combin. Designs 12 (2004) 197–220; A classification result on weighted image-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 image, with image the size of the smallest non-trivial blocking sets in image. In this article, we improve this latter result for weighted image-minihypers, without restrictions on the weights of the points. The largest improvements are obtained for image, p prime, image, where we increase the upper bound to image.