Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth

Katz, Daniel J.

doi:10.1016/j.jcta.2012.05.003

Cited by 29 publications

(36 citation statements)

References 19 publications

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“…The formula in (7) was already observed by Katz [38]. However, our discussion in this section may give more information on the coding theory problem and the sequence problem, as the formula of (6) is new.…”

Section: Weights In Cmentioning

confidence: 65%

Three-weight cyclic codes and their weight distributions

Ding

et al. 2016

Discrete Mathematics

115

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a b s t r a c tCyclic codes have been an important topic of both mathematics and engineering for decades. They have been widely used in consumer electronics, data transmission technologies, broadcast systems, and computer applications as they have efficient encoding and decoding algorithms. The objective of this paper is to provide a survey of three-weight cyclic codes and their weight distributions. Information about the duals of these codes is also given when it is available.

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Section: Weights In Cmentioning

confidence: 65%

Three-weight cyclic codes and their weight distributions

Ding

et al. 2016

Discrete Mathematics

115

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“…Our Weil sums in (1) are also of practical interest, as they determine the performance of protocols in communications theory, remote sensing, cryptography, and coding theory. See the Appendix of [26] for how these sums relate to correlation of sequences and nonlinearity of boolean functions.…”

Section: Introductionmentioning

confidence: 99%

“…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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“…It was conjectured by Helleseth [12] that in these cases, any decimation gives at least four crosscorrelation values. This conjecture has recently been proved in the binary case by Katz [20]. The general case to settle the conjecture for all other values of p is still open.…”

Section: Theorem 5 Let P Be An Odd Prime Then the Following Decimatimentioning

confidence: 93%

Open Problems on the Cross-correlation of m-Sequences

Helleseth

2014

Open Problems in Mathematics and Computational Science

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Pseudorandom sequences are important for many applications in communication systems, in coding theory, and in the design of stream ciphers. Maximumlength linear sequences (or m-sequences) are popular in sequence designs due to their long period and excellent pseudorandom properties. In code-division multiple-access (CDMA) applications, there is a demand for large families of sequences having good correlation properties. The best families of sequences in these applications frequently use m-sequences in their constructions. Therefore, the problem of determining the correlation properties of m-sequences has received a lot of attention since the 1960s, and many interesting theoretical results of practical interest have been obtained. The cross-correlation of m-sequences is also related to other important problems, such as almost perfect nonlinear functions (APN) and almost bent functions (AB), and to the nonlinearity of S-boxes in many block ciphers including AES. This chapter gives an updated survey of the cross-correlation of m-sequences and describes some of the most important open problems that still remain in this area.

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Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth

Cited by 29 publications

References 19 publications

Three-weight cyclic codes and their weight distributions

Three-weight cyclic codes and their weight distributions

The p-adic valuations of Weil sums of binomials

Open Problems on the Cross-correlation of m-Sequences

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