Cyclotomy of Weil sums of binomials

Aubry, Yves; Katz, Daniel J.; Langevin, Philippe

doi:10.1016/j.jnt.2015.02.011

Cited by 13 publications

(19 citation statements)

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“…Since W F,d (a) always lies in Q(e 2πi/p ), we see that its valuation must be an integer multiple of 1/(p−1). Bounds on the p-adic valuation of W F,d (a) have proved very helpful in determining the values of W F,d (a), as can be seen in [1,2,3,4,5,6,7,10,15,16,20,21,24,26,27,28,29,34,36,37,40,41]. The main tool in determining the p-adic valuation of these Weil sums is Stickelberger's Theorem on the valuation of Gauss sums, which allows for an exact determination of (2) V F,d = min a∈F val p (W F,d (a)) in terms of a combinatorial formula that is given in Lemma 2.9 below.…”

Section: Introductionmentioning

confidence: 99%

The p-adic valuations of Weil sums of binomials

Katz

Langevin

Lee

et al. 2017

Journal of Number Theory

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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 .

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Section: Introductionmentioning

confidence: 99%

The p-adic valuations of Weil sums of binomials

Katz

Langevin

Lee

et al. 2017

Journal of Number Theory

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“…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].…”

Section: Non-archimedean Boundsmentioning

confidence: 99%

“…This conjecture has been studied by many researchers [5,55,13,4,24,43,1,2,44,46], and was proved to be true in characteristics 2 and 3 (see [ A corollary of this theorem is that if there is any counterexample to Conjecture 10.6, then it cannot be symmetric. Corollary 10.9.…”

Section: Number Of Valuesmentioning

confidence: 99%

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

Katz¹

2019

Combinatorics and Finite Fields

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We present a survey on Weil sums in which an additive character of a finite field F is applied to a binomial whose individual terms (monomials) become permutations of F when regarded as functions. Then we indicate how these Weil sums are used in applications, especially how they characterize the nonlinearity of power permutations and the correlation of linear recursive sequences over finite fields. In these applications, one is interested in the spectrum of Weil sum values that are obtained as the coefficients in the binomial are varied. We review the basic properties of such spectra, and then give a survey of current topics of research: Archimedean and non-Archimedean bounds on the sums, the number of values in the spectrum, and the presence or absence of zero in the spectrum. We indicate some important open problems and discuss progress that has been made on them.

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“…Here we only consider two cases. Table II which are listed in [13], where q = 2 n and 1 ≤ d ≤ q − 1. We refer the reader to [13] and the references therein for detailed information.…”

Section: B Three Weights Linear Codes Over F 2 Tmentioning

confidence: 99%

A Construction of Linear Codes Over ${\mathbb {F}}_{2^t}$ From Boolean Functions

Xiang

Feng

Tang

2017

IEEE Trans. Inform. Theory

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In this paper, we present a construction of linear codes over F 2 t from Boolean functions, which is a generalization of Ding's method. Based on this construction, we give two classes of linear codesC f and C f over F 2 t from a Boolean function f : F q → F 2 , where q = 2 n and F 2 t is some subfield of F q . The complete weight enumerator ofC f can be easily determined from the Walsh spectrum of f , while the weight distribution of the code C f can also be easily settled. Particularly, the number of nonzero weights ofC f and C f is the same as the number of distinct Walsh values of f . As applications of this construction, we show several series of linear codes over F 2 t with two or three weights by using bent, semibent, monomial and quadratic Boolean function f . Linear codes, Walsh spectrum, Boolean function, bent and semibent functions, weight enumerator.

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Cyclotomy of Weil sums of binomials

Cited by 13 publications

References 29 publications

The p-adic valuations of Weil sums of binomials

The p-adic valuations of Weil sums of binomials

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

A Construction of Linear Codes Over ${\mathbb {F}}_{2^t}$ From Boolean Functions

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