Power sums over subspaces of finite fields (Q439087)

Let \(F\) be the finite field of order \(q\), a prime power, and let \(K\) be an extension of degree \(m+1\). For an \(F\)-subspace \(V\) of \(K\) and \(\alpha\in K\), define NEWLINE\[NEWLINE S_t(V, \alpha )=\sum_{v\in V} (v+\alpha)^t. NEWLINE\]NEWLINE Such sums have arisen in the study of cyclic codes and the study of Galois modules of \(p\)-adic fields. When \(q\) is a prime and \(\dim V=m\) then \textit{N. P. Byott} and \textit{R. J. Chapman} [Finite Fields Appl. 5, No. 3, 254--265 (1999; Zbl 1029.11069)] determined the \(t\) where \(S_t(V \alpha )\) is zero. Here this result is extended to arbitrary \(q\) but using a completely different approach. They take a particular expansion of \(S_t\) and apply some coding theory results.