Parameters for which the Griesmer bound is not sharp

We prove for a large class of parameters t and r that a multiset of at most tθ_d-k+rθ_d-k-2 points in PG(d,q) that blocks every k-dimensional subspace at least t times must contain a sum of t subspaces of codimension k.We use our results to identify a class of parameters for linear codes for which the Griesmer bound is not sharp. Our theorem generalizes the non-existence results from Maruta On the achievement of the Griesmer bound, Des. Codes Cryptogr. 12 (1997) 83-87] and Klein On codes meeting the Griesmer bound, Discrete Math. 274 (2004) 289-297].