# 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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topMarko 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

top- [1] R. S. Coulter, Explicit evaluations of some Weil sums, Acta Arith. 83 (1998), 241-251. Zbl0924.11098
- [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] R. J. McEliece, Finite Fields for Computer Scientists and Engineers, Kluwer, Dordrecht, 1987. Zbl0662.94014
- [4] M. J. Moisio, On relations between certain exponential sums and multiple Kloosterman sums and some applications to coding theory, preprint, 1997.
- [5] M. J. Moisio, Exponential sums, Gauss sums and cyclic codes, Dissertation, Acta Univ. Oul. A 306, 1998. Zbl0970.94011

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