Weighted [formula omitted]-minihypers

Abstract

In Ferret and Storme [A classification result on weighted { δ ( p 3 + 1 ) , δ ; 3 , p 3 } -minihypers, J. Combin. Designs 12 (2004) 197–220; A classification result on weighted { δ v μ + 1 , δ v μ ; N , p 3 } -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 { δ v μ + 1 , δ v μ ; 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 ( p 3 + 1 ) , δ ; 3 , p 3 } -minihypers, J. Combin. Designs 12 (2004) 197–220; A classification result on weighted { δ v μ + 1 , δ v μ ; N , p 3 } -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 ⩽ ε , with q + 1 + ε the size of the smallest non-trivial blocking sets in PG ( 2 , q ) . In this article, we improve this latter result for weighted { δ ( q + 1 ) , δ ; N , q } -minihypers, without restrictions on the weights of the points. The largest improvements are obtained for q = p 2 , p prime, p ⩾ 11 , where we increase the upper bound to δ ⩽ ( q - 1 ) / 4 .