For q = p m with p prime and k | q − 1, we consider the generalized Paley graphq }, and the irreducible p-ary cyclic code C(k, q) = {(Tr q/p (γω ik ) n−1 i=0 )} γ∈Fq , with ω a primitive element of Fq and n = q−1 k . 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−1 p−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.
We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set Fqm and connection set the nonzero (q ℓ + 1)-th powers in Fqm , as well as their complements. We explicitly compute the spectrum of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular (of Latin square type in half of the cases). As applications, on the one hand we solve Waring's problem over Fqm for the exponents q ℓ +1, for each q and for infinite values of ℓ and m. We obtain that the Waring's number g(q ℓ + 1, q m ) = 1 or 2, depending on m and ℓ, thus tackling some open cases. On the other hand, we construct infinite towers of Ramanujan graphs in all characteristics.
We show that the Waring's number over a finite field Fq, denoted g(k, q), when exists, coincides with the diameter of the generalized Paley graphWe find infinite new families of exact values of g(k, q) from a characterization of graphs Γ(k, q) which are also Hamming graphs proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring's number for some pair (k, q) with q not a prime. Finally, we find a lower bound for g(k, p) with p prime by using that Γ(k, p) is a circulant graph in this case.The goal of the paper is to give some new exact values for g(k, p m ) and a lower bound for g(k, p), where p is a prime.
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