# The spectra of generalized Paley graphs and their associated irreducible cyclic codes

For q=p^m with p prime and k| q-1, we consider the generalized Paley graph Γ(k,q) = Cay(F_q, R_k), with R_k={ x^k : x ∈F_q^* }, and the irreducible p-ary cyclic code C(k,q) = {(Tr_q/p(γω^ik)_i=0^n-1)}_γ∈F_q, with ω a primitive element of F_q and n=q-1k. We compute the spectra of Γ(k,q) in terms of Gaussian periods and give Spec(Γ(k,q)) explicitly in the semiprimitive case. We then show that the spectra of Γ(k,q) and C(k,q) are mutually determined by each other if further k|q-1p-1. Also, we use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves and to the computation of Gaussian periods.

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