# A note on evaluations of some exponential sums

Acta Arithmetica (2000)

• Volume: 93, Issue: 2, page 117-119
• ISSN: 0065-1036

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## Abstract

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1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S\left(a,{p}^{\alpha }+1\right):={\sum }_{x{\in }_{q}}\chi \left(a{x}^{{p}^{\alpha }+1}\right)$ where χ is a non-trivial additive character of the finite field ${}_{q}$, $q={p}^{e}$ odd, and $a\in {*}_{q}$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of ${p}^{\alpha }+1$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain sums of the form ${\sum }_{x{\in }_{q}}\chi \left(a{x}^{{p}^{\alpha }+1}+bx\right)$.

## How to cite

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Marko J. Moisio. "A note on evaluations of some exponential sums." Acta Arithmetica 93.2 (2000): 117-119. <http://eudml.org/doc/207403>.

@article{MarkoJ2000,
abstract = {1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S(a,p^\{α\}+1) := ∑_\{x∈_q\} χ(ax^\{p^\{α\}+1\})$ where χ is a non-trivial additive character of the finite field $_q$, $q = p^e$ odd, and $a ∈ *_q$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of $p^\{α\}+1$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain sums of the form $∑_\{x∈_q\} χ(ax^\{p^\{α\}+1\} + bx)$.},
author = {Marko J. Moisio},
journal = {Acta Arithmetica},
keywords = {finite fields; Gauss sums; Weil sums; Davenport-Hasse relations},
language = {eng},
number = {2},
pages = {117-119},
title = {A note on evaluations of some exponential sums},
url = {http://eudml.org/doc/207403},
volume = {93},
year = {2000},
}

TY - JOUR
AU - Marko J. Moisio
TI - A note on evaluations of some exponential sums
JO - Acta Arithmetica
PY - 2000
VL - 93
IS - 2
SP - 117
EP - 119
AB - 1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S(a,p^{α}+1) := ∑_{x∈_q} χ(ax^{p^{α}+1})$ where χ is a non-trivial additive character of the finite field $_q$, $q = p^e$ odd, and $a ∈ *_q$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of $p^{α}+1$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain sums of the form $∑_{x∈_q} χ(ax^{p^{α}+1} + bx)$.
LA - eng
KW - finite fields; Gauss sums; Weil sums; Davenport-Hasse relations
UR - http://eudml.org/doc/207403
ER -

## References

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1. [1] R. S. Coulter, Explicit evaluations of some Weil sums, Acta Arith. 83 (1998), 241-251. Zbl0924.11098
2. [2] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia Math. Appl. 20, Addison-Wesley, Reading, 1983 (now distributed by Cambridge Univ. Press). Zbl0554.12010
3. [3] R. J. McEliece, Finite Fields for Computer Scientists and Engineers, Kluwer, Dordrecht, 1987. Zbl0662.94014
4. [4] M. J. Moisio, On relations between certain exponential sums and multiple Kloosterman sums and some applications to coding theory, preprint, 1997.
5. [5] M. J. Moisio, Exponential sums, Gauss sums and cyclic codes, Dissertation, Acta Univ. Oul. A 306, 1998. Zbl0970.94011

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