The p-adic valuations of Weil sums of binomials

Abstract

Abstract. We investigate the p-adic valuation of Weil sums of the form W F,d (a) = x∈F ψ(x d − ax), where F is a finite field of characteristic p, ψ is the canonical additive character of F , the exponent d is relatively prime to |F × |, and a is an element of F . Such sums often arise in arithmetical calculations and also have applications in information theory. For each F and d one would like to know V F,d , the minimum p-adic valuation of W F,d (a) as a runs through the elements of F . We exclude exponents d that are congruent to a power of p modulo |F × | (degenerate d), which yield trivial Weil sums. We prove that V F,d ≤ (2/3)[F : Fp] for any F and any nondegenerate d, and prove that this bound is actually reached in infinitely many fields F . We also prove some stronger bounds that apply when [F : Fp] is a power of 2 or when d is not congruent to 1 modulo p−1, and show that each of these bounds is reached for infinitely many F .

7. Weil sums of binomials: properties, applications and open problems

“…(But also see [51,52] for approaches proceeding directly from Stickelberger's theorem in characteristic 2.) The more general result in the lemma here is stated and proved as [47,Lemma 2.9]. The same result also appears as [45,Proposition 4.3], with the |F | > 2 condition missing, which does not affect the other results of that paper.…”

“…Corollary 9.3 tells us how to obtain the valuation of the Weil spectrum from those of pairwise products of Gauss sums. Now Stickelberger's Theorem tells us the p-adic valuation of these Gauss sums, which in turn gives the exact value of V F,d in terms of a combinatorial problem concerning quantities 2 And for the same reason, the formulae in [47,Lemma 2.5] should have the same sign changes, but one can see that this does not affect the proof of Corollary 2.6 of that paper, which is the only place this result is used.…”

“…And for the same reason, the formulae in[47, Lemma 2.5] should have the same sign changes, but one can see that this does not affect the proof of Corollary 2.6 of that paper, which is the only place this result is used.3 Also see [2, Lemma 4.1], which neglects the fact that the trivial case when |F | = 2 must be handled separately; the rest of that paper is unaffected by this oversight because the lemma is used only in Corollary 4.2 of that paper, which remains true because V L,d ≤ [L : F2] can be deduced immediately from the fact that the first power moment for W L,d (given in Lemma 2.1(i) of that paper) has 2-adic valuation [L : F2].…”

“…. In many cases, the upper bounds of Theorem 9.7 are the best possible, as discussed in [47,Remark 1.3]. If we are not concerned about the degeneracy of the exponent over the prime field, we get the following simplified version of Theorem 9.7. ]…”

“…See [47,Corollary 2.6] for a proof. When d ≡ 1 (mod p − 1) (which is invariably true in characteristic 2), one can obtain an equivalent result from McEliece's Theorem [53,54] on p-divisibility of weights in cyclic codes; earlier uses of this principle in characteristic 2 often proceeded in this way [56,5,6,7,8,37,38,22].…”