Orthogonal systems of polynomials in finite fields

Author:
H. Niederreiter

Journal:
Proc. Amer. Math. Soc. **28** (1971), 415-422

MSC:
Primary 12C05

DOI:
https://doi.org/10.1090/S0002-9939-1971-0291136-9

MathSciNet review:
0291136

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Abstract: The notion of an orthogonal system of polynomials in several variables in finite fields is introduced which generalizes a concept of orthogonality by Kurbatov and Starkov. Necessary and sufficient conditions for orthogonality in terms of character sums and permutation polynomials are given. Results of Carlitz on systems of equations in finite fields and earlier results of the author on permutation polynomials in several variables are generalized.

**[1]**L. Carlitz,*Invariantive theory of equations in a finite field*, Trans. Amer. Math. Soc.**75**(1953), 405-127. MR**15**, 291. MR**0057912 (15:291c)****[2]**-,*Invariant theory of systems of equations in a finite field*, J. Analyse Math.**[3]**L. E. Dickson,*General theory of modular invariants*, Trans. Amer. Math. Soc.**10**(1909), 123-158. MR**1500831****[4]**V. A. Kurbatov and N. G. Starkov,*The analytic representation of permutations*, Sverdlovsk. Gos. Ped. Inst. Učen. Zap.**31**(1965), 151-158. (Russian) MR**35**#6652. MR**0215817 (35:6652)****[5]**H. Niederreiter,*Permutation polynomials in several variables over finite fields*, Proc. Japan Acad.**46**(1970), 1001-1005. MR**0288100 (44:5298)****[6]**H. Niederreiter,*Permutation polynomials in several variables*, Acta. Sci. Math. (Szeged) (to appear). MR**0309894 (46:8998)**

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DOI:
https://doi.org/10.1090/S0002-9939-1971-0291136-9

Keywords:
Orthogonal systems of polynomials,
permutation polynomials,
equations in finite fields

Article copyright:
© Copyright 1971
American Mathematical Society