## Incomplete additive character sums and applications.(English)Zbl 1019.11034

Jungnickel, Dieter (ed.) et al., Finite fields and applications. Proceedings of the fifth international conference on finite fields and applications $$F_q5$$, University of Augsburg, Germany, August 2-6, 1999. Berlin: Springer. 462-474 (2001).
Let $$F_{q}$$ be a finite field, $$S$$ a subset of $$F_{q}$$, $$\chi$$ a multiplicative character, $$\psi$$ the canonical additive character of $$F_{q}$$, and $T(g,S)= \sum_{x \in S} \psi(g(x)), \qquad W(f,g,S)= \sum_{x \in S} \chi(f(x)) \psi(g(x))$ incomplete character sums of polynomials $$f,g \in F_{q}[x]$$. The author proves that for any $$S \subset F_{q}$$ and $$n \geq 2$$ with $$(n,q-1)$$ there exists an irreducible polynomial $$g \in F_{q}[x]$$ of degree $$n$$ such that $|T(g,S)|\geq \sqrt{|S|\left( 1+{|S|-1 \over (q-1)^{2}} \right)}.$ Similarly, for any $$S \subset F_{q}$$, $$m \geq 2$$ with $$\text{ord} (\chi) \nmid m$$, and any $$g \in F_{q}[x]$$ there exists a monic irreducible polynomial $$f \in F_{q}[x]$$ of degree $$m$$ such that $|W(f,g,S)|\geq \sqrt{|S|}$ [see also A. Winterhof, Des. Codes Cryptography 22, 123-131 (2001; Zbl 0995.11067)] and S. A. Stepanov [Arithmetic of algebraic curves, Plenum, New York (1994; Zbl 0862.11036), pp. 85-86].
The author gives also a series of new upper bounds for the above character sums. Let $$q=p^{\nu}$$ where $$p$$ is a prime, $$\{\omega_{0}, \ldots, \omega_{\nu-1} \}$$ be a basis of $$F_{q}$$ over $$F_{p}$$, and $$g \in F_{q}[x]$$ a polynomial of degree $$n$$ with $$(n,q)=1$$. If $B= \{x=x_{0} \omega_{0}+ \cdots +x_{\nu-1} \omega_{\nu-1} \mid 0 \leq x_{i} < k_{i} \leq p, 0 \leq i \leq \nu-1 \}$ and $T(g,B)= \sum_{x \in B} \psi(g(x)),$ the author proves that $|T(g,B)|\leq(n-1)q^{1/2}(1+ \log p)^{\nu}.$ This extends the corresponding results of Vinogradov, Polya, Davenport and Lewis for linear polynomials, and a result of Burgess for arbitrary polynomials.
Now if $$i= i_{0}+ i_{1}p+ \cdots + i_{\nu-1}p^{\nu-1}$$ is a $$p$$-adic representation of a positive integer $$i$$, let $S= \{x_{i}= i_{0} \omega_{0}+ \cdots +i_{\nu-1} \omega_{\nu-1} \mid 0 \leq i \leq k-1 \}$ and $T(g, S)= \sum_{x_{i} \in S} \psi(g(x_{i})).$ The author proves that $|T(g, S)|\leq (n-1)q^{1/2}(1+ \log q).$ Moreover, he shows that $|\sum_{x \in A} \psi(g(x))|\leq (n-1)q^{1/2},$ for any additive subgroup $$A$$ of $$F_{q}$$. Similar results are obtained for multiplicative character sums $U(f,B)= \sum_{x \in B} \chi(f(x)), \quad U(f,S)= \sum_{x_{i} \in S} \chi(f(x_{i})), \quad U(f,A)= \sum_{x \in A} \chi(f(x))$ and for hybrid sums $W(f,g,B)= \sum_{x \in B} \chi(f(x)) \psi(g(x)),\quad W(f,g,S)= \sum_{x_{i} \in S} \chi(f(x_{i})) \psi(g(x_{i}))$ and $W(f,g,A)= \sum_{x \in A} \chi(f(x)) \psi(g(x)).$
The above results are applied to a variant of Waring’s problem in finite fields, the distribution properties of irreducible polynomials in $$F_{q}[x]$$, and the study of dc-constrained codes with additive characters.
For the entire collection see [Zbl 0959.00027].

### MSC:

 11T23 Exponential sums 11L07 Estimates on exponential sums 94B15 Cyclic codes 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)

### Keywords:

finite fields; character sums and dc-constrained codes

### Citations:

Zbl 0995.11067; Zbl 0862.11036