Cyclotomic numbers of order 5 over F/sub p//sup n/

In this paper, we derive the cyclotomic numbers of order 5 over an extension field F_p ⁿ using the well-known results of quintic Jacobi sums over F_p (B. C. Berndt, et al., 1998). For p ne 1 mod 5, we have obtained the simple closed-form expression of the cyclotomic numbers of order 5 over F_p ⁿ. For p equiv 1 mod 5, we express the cyclotomic number of order 5 over F _p ⁿ in terms of the solution of the diophantine system which is required to evaluate the cyclotomic number of order 5 over F_p ⁿ. Using the cyclotomic numbers of order 5 over F_p ⁿ, autocorrelation distributions of 5-ary Sidel´nikov sequences of period pⁿ - 1 are also derived