Divisibility of Weil sums of binomials
HTML articles powered by AMS MathViewer
- by Daniel J. Katz
- Proc. Amer. Math. Soc. 143 (2015), 4623-4632
- DOI: https://doi.org/10.1090/proc/12687
- Published electronically: April 1, 2015
- PDF | Request permission
Abstract:
Consider the Weil sum $W_{F,d}(u)=\sum _{x \in F} \psi (x^d+u x)$, where $F$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $F$, $d$ is coprime to $|F^*|$, and $u \in F^*$. We say that $W_{F,d}(u)$ is three-valued when it assumes precisely three distinct values as $u$ runs through $F^*$: this is the minimum number of distinct values in the nondegenerate case, and three-valued $W_{F,d}$ are rare and desirable. When $W_{F,d}$ is three-valued, we give a lower bound on the $p$-adic valuation of the values. This enables us to prove the characteristic $3$ case of a 1976 conjecture of Helleseth: when $p=3$ and $[F:{\mathbb F}_3]$ is a power of $2$, we show that $W_{F,d}$ cannot be three-valued.References
- Yves Aubry, Daniel J. Katz, and Philippe Langevin, Cyclotomy of Weil sums of binomials, arXiv:1312.3889 (2013).
- Emrah ÇakÇak and Philippe Langevin, Power permutations in dimension 32, Sequences and their applications—SETA 2010, Lecture Notes in Comput. Sci., vol. 6338, Springer, Berlin, 2010, pp. 181–187. MR 2830722, DOI 10.1007/978-3-642-15874-2_{1}4
- A. R. Calderbank, Gary McGuire, Bjorn Poonen, and Michael Rubinstein, On a conjecture of Helleseth regarding pairs of binary $m$-sequences, IEEE Trans. Inform. Theory 42 (1996), no. 3, 988–990. MR 1445885, DOI 10.1109/18.490561
- L. Carlitz, A note on exponential sums, Math. Scand. 42 (1978), no. 1, 39–48. MR 500144, DOI 10.7146/math.scand.a-11734
- L. Carlitz, Explicit evaluation of certain exponential sums, Math. Scand. 44 (1979), no. 1, 5–16. MR 544577, DOI 10.7146/math.scand.a-11793
- Pascale Charpin, Cyclic codes with few weights and Niho exponents, J. Combin. Theory Ser. A 108 (2004), no. 2, 247–259. MR 2098843, DOI 10.1016/j.jcta.2004.07.001
- Todd Cochrane and Christopher Pinner, Stepanov’s method applied to binomial exponential sums, Q. J. Math. 54 (2003), no. 3, 243–255. MR 2013138, DOI 10.1093/qjmath/54.3.243
- Todd Cochrane and Christopher Pinner, Explicit bounds on monomial and binomial exponential sums, Q. J. Math. 62 (2011), no. 2, 323–349. MR 2805207, DOI 10.1093/qmath/hap041
- Robert S. Coulter, Further evaluations of Weil sums, Acta Arith. 86 (1998), no. 3, 217–226. MR 1655980, DOI 10.4064/aa-86-3-217-226
- H. Davenport and H. Heilbronn, On an Exponential Sum, Proc. London Math. Soc. (2) 41 (1936), no. 6, 449–453. MR 1576618, DOI 10.1112/plms/s2-41.6.449
- Tao Feng, On cyclic codes of length $2^{2^r}-1$ with two zeros whose dual codes have three weights, Des. Codes Cryptogr. 62 (2012), no. 3, 253–258. MR 2886276, DOI 10.1007/s10623-011-9514-0
- Richard A. Games, The geometry of $m$-sequences: three-valued crosscorrelations and quadrics in finite projective geometry, SIAM J. Algebraic Discrete Methods 7 (1986), no. 1, 43–52. MR 819704, DOI 10.1137/0607005
- Tor Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math. 16 (1976), no. 3, 209–232. MR 429323, DOI 10.1016/0012-365X(76)90100-X
- A. A. Karacuba, Estimates of complete trigonometric sums, Mat. Zametki 1 (1967), 199–208 (Russian). MR 205941
- Daniel J. Katz, Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth, J. Combin. Theory Ser. A 119 (2012), no. 8, 1644–1659. MR 2946379, DOI 10.1016/j.jcta.2012.05.003
- Nicholas Katz and Ron Livné, Sommes de Kloosterman et courbes elliptiques universelles en caractéristiques $2$ et $3$, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 11, 723–726 (French, with English summary). MR 1054286
- H. D. Kloosterman, On the representation of numbers in the form $ax^2+by^2+cz^2+dt^2$, Acta Math. 49 (1927), no. 3-4, 407–464. MR 1555249, DOI 10.1007/BF02564120
- Gilles Lachaud and Jacques Wolfmann, Sommes de Kloosterman, courbes elliptiques et codes cycliques en caractéristique $2$, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 20, 881–883 (French, with English summary). MR 925289
- Rudolf Lidl and Harald Niederreiter, Finite fields, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. MR 1429394
- Gary McGuire, On certain 3-weight cyclic codes having symmetric weights and a conjecture of Helleseth, Sequences and their applications (Bergen, 2001) Discrete Math. Theor. Comput. Sci. (Lond.), Springer, London, 2002, pp. 281–295. MR 1916139
- I. Vinogradov, Some trigonometrical polynomes and their applications, C. R. Acad. Sci. URSS (N.S.) (1933), no. 6, 254–255.
Bibliographic Information
- Daniel J. Katz
- Affiliation: Department of Mathematics, California State University, Northridge, California 91330-8313
- MR Author ID: 787969
- Received by editor(s): July 29, 2014
- Published electronically: April 1, 2015
- Communicated by: Matthew A. Papanikolas
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4623-4632
- MSC (2010): Primary 11T23, 11L05, 11L07; Secondary 11T71
- DOI: https://doi.org/10.1090/proc/12687
- MathSciNet review: 3391022