Power sums over subspaces of finite fields

Let K be the finite field of order qm+1, which is regarded as an (m+1)-dimensional vector space over Fq. For each h-dimensional Fq-subspace V of K, α∈K and 0⩽t⩽qm+1−1, we define St(V,α)=∑v∈V(α+v)t. For each 1⩽h⩽m, we obtain sufficient conditions on t for the vanishing of St(V,α); when h=m, combining this result with some p-rank results from coding theory, we obtain necessary and sufficient conditions on t for the vanishing of St(V,α).