Yingjie CHENG, Xiwang CAO, Liqin QIAN, Jinlong WAN

Estimations on some hybrid exponential sums related to Kloosterman sums

In this paper, we revisit the bounds of the mixed exponential sums introduced by Lv and Zhang (2020).Moreover, we give some estimations for some new hybrid exponential sums related to Kloosterman sums over finite fieldsof odd characteristic by using the properties of Jacobi sums and Gaussian sums.

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[1] Bourgain J. Estimates on exponential sums related to the Diffie–Hellman Distributions. Geometric and Functional Analysis 2005; 15 (1): 1-34. doi: 10.1007/s00039-005-0500-4
[2] Coulter RS. Further evaluations of some Weil sums. Acta Arithmetica 1998; 86 (3): 217-226. doi: 10.4064/aa-86-3- 217-226
[3] Di H. A hybrid mean value involving two-term exponential sums and polynomial character sums. Czechoslovak Mathematical Journal 2014; 64 (139): 53–62. doi: 10.1007/s10587-014-0082-0
[4] Hurt NE. Exponential sums and coding theory: a review. Acta Applicandae Mathematicae 1997; 46 (1): 49-91. doi: 10.1023/A:1005794417363
[5] Luo GJ, Cao XW. Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum. IEEE Transactions on Information Theory 2018; 64 (10): 6948-6505. doi: 10.1109/TIT.2017.2777492
[6] Luo GJ, Cao XW. A construction of linear codes and strongly regular graphs from q-polynomials. Discrete Mathematics 2017; 340: 2262-2274. doi: 10.1016/j.disc.2017.04.016
[7] Liu WM, Li XY. A hybrid mean value involving the two-term exponential sums and two-term character sums. Journal of Applied Mathematics 2014; 5: 1-5. doi: 10.1155/2014/845845
[8] Lidl R, Niederreiter H. Finite Fields. In: Encyclopedia of Mathematics and its Applications. 2nd ed. Cambridge, UK:Cambridge University Press, 1997.
[9] Li CJ, Wu P, Liu FM. On two classes of primitive BCH codes and some related codes. IEEE Transactions on Information Theory 2018; 65 (6): 3830-3840. doi: 10.1109/TIT.2018.2883615
[10] Lv XX, Zhang WP. A new hybrid power mean involving the generalized quadratic Gauss sums and sums analogous to Kloosterman sums. Lithuanian Mathematical Journal 2017; 57 (3): 359-366. doi: 10.1007/s10986-017-9366-z
[11] Lv XX, Zhang WP. On the character sum of Polynomials and the two-term exponential sums. Acta Mathematica Sinica, English Series 2020; 36 (2): 196-206. doi: 10.1007/s10114-020-9255-y
[12] Paterson KG. Application of exponential sums in communications theory. In: Walker M (editor). Cryptography and Coding. Lecture Notes in Computer Science, Vol. 1746. Springer, Berlin, Germany: Springer, 1999. doi: 10.1007/3- 540-46665-7_1
[13] Wu YN, Li N, Zeng XY. Linear codes with few weights from cyclotomic classes and weakly regular bent fuctions. Designs Codes and Cryptography 2020; 88 (4-5): 1255-1272. doi: 10.1007/s10623-020-00744-9
[14] Xu ZF, Zhang TP, Zhang WP. On the mean value of the two-term exponential sums with Dirichlet characters. Journal of Number Theory 2007; 123 (2): 352-362. doi: 10.1016/j.jnt.2006.07.005
[15] Zhang H, Zhang WP. The fourth power mean of two-term exponential sums and its application. Mathematical Reports 2017; 19 (69): 75-81. doi: 10.1155/2014/724840
[16] Zhang WP. The fourth and sixth power mean of the classical Kloosterman sums. Journal of Number Theory 2011; 131 (2): 228-238. doi: 10.1016/j.jnt.2010.08.008
[17] Zhang WP, Han D. A new identity involving the classical Kloosterman sums and 2-dimensional Kloosterman sums. International Journal of Number Theory 2016; 12 (1): 111–119. doi: 10.1142/S179304211650007X