Upper bounds on cyclotomic numbers

In this article, we give upper bounds for cyclotomic numbers of order e over a finite field with q elements, where e is a divisor of q-1. In particular, we show that under certain assumptions, cyclotomic numbers are at most $\lceil \frac{k}{2}\rceil$, and the cyclotomic number (0,0) is at most $\lceil \frac{k}{2}\rceil -1$, where k=(q-1)/e. These results are obtained by using a known formula for the determinant of a matrix whose entries are binomial coefficients.