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 positive divisor of q − 1. In particular, we show that under certain assumptions, cyclotomic numbers are at most ⌈k2⌉, and the cyclotomic number (0, 0) is at most ⌈k2⌉-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.