Excluding three correlation levels in quadratic field towers
Let $F$ be a finite field of characteristic $p$ and $d$ a positive integer coprime to $|F| - 1$. For $a \in F$ let $$W_{F,d}(a) = \sum_{x \in F} \exp\Bigl(\frac{2\pi i}{p} \operatorname{Tr}_{F/\mathbb F_p}\bigl(x^d - a x\bigr)\Bigr).$$ The values $-1 + W_{F,d}(a)$, $a \ne 0$, are the cross-correlation values of a $p$-ary $m$-sequence of period $|F| - 1$ and its decimation by $d$. If $d$ is congruent to a power of $p$ modulo $|F| - 1$, the set $\{W_{F,d}(a) : a \in F \setminus \{0\}\}$ has at most two elements; otherwise it has at least three (Helleseth). Three-element sets occur for some $d$ whenever $[F : \mathbb F_p]$ is not a power of $2$, and Helleseth conjectured that they never occur when it is. Katz proved this in characteristics $2$ (completing work of Feng) and $3$. Katz also showed that a three-element set always contains $0$, and Aubry, Katz and Langevin showed that when $[F : \mathbb F_p]$ is a power of $2$ its two nonzero elements cannot be negatives of each other.
Problem. Let $p \ge 5$ be prime, $r \ge 0$ an integer, and $F = \mathbb F_{p^{2^r}}$. Must the set $$\{W_{F,d}(a) : a \in F \setminus \{0\}\}$$ have cardinality different from three, for every positive integer $d$ coprime to $|F| - 1$? No condition $d \equiv 1 \pmod{p-1}$ is imposed.
Source: Helleseth (1971, master's thesis; 1976, Conjecture 5.2), the three-valued conjecture; see Katz (2019), Conjecture 10.6. Stated for characteristic p ≥ 5 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NT — Number Theory
- Secondary
- CO, IT
- Keywords
- AIM Open Applied Problems, finite fields, m-sequences, Weil sums
References
@mastersthesis{helleseth1971,
author = {Helleseth, Tor},
title = {Krysskorrelasjonsfunksjonen mellom maksimale sekvenser over {GF}(q)},
school = {Universitetet i Bergen},
year = {1971},
}
@article{helleseth1976,
author = {Helleseth, Tor},
title = {Some results about the cross-correlation function between two maximal linear sequences},
journal = {Discrete Math.},
volume = {16},
number = {3},
year = {1976},
pages = {209--232},
doi = {10.1016/0012-365X(76)90100-X},
}
@incollection{katz2019,
author = {Katz, Daniel J.},
title = {Weil sums of binomials: properties, applications and open problems},
booktitle = {Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications},
editor = {Schmidt, Kai-Uwe and Winterhof, Arne},
series = {Radon Ser. Comput. Appl. Math.},
volume = {23},
publisher = {De Gruyter},
address = {Berlin},
year = {2019},
pages = {109--134},
doi = {10.1515/9783110642094-007},
eprint = {1805.10452},
archivePrefix = {arXiv},
}
@article{feng2012,
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}
@article{katz2012,
author = {Katz, Daniel J.},
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journal = {J. Combin. Theory Ser. A},
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number = {8},
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doi = {10.1016/j.jcta.2012.05.003},
eprint = {1105.2291},
archivePrefix = {arXiv},
}
@article{katz2015,
author = {Katz, Daniel J.},
title = {Divisibility of {W}eil sums of binomials},
journal = {Proc. Amer. Math. Soc.},
volume = {143},
number = {11},
year = {2015},
pages = {4623--4632},
doi = {10.1090/proc/12687},
eprint = {1407.7923},
archivePrefix = {arXiv},
}
@article{aubry2015,
author = {Aubry, Yves and Katz, Daniel J. and Langevin, Philippe},
title = {Cyclotomy of {W}eil sums of binomials},
journal = {J. Number Theory},
volume = {154},
year = {2015},
pages = {160--178},
doi = {10.1016/j.jnt.2015.02.011},
eprint = {1312.3889},
archivePrefix = {arXiv},
}
@article{nguyen2022,
author = {Nguyen, Liem},
title = {On {W}eil sums, conjectures of {H}elleseth, and {N}iho exponents},
journal = {J. Number Theory},
volume = {234},
year = {2022},
pages = {240--258},
doi = {10.1016/j.jnt.2021.06.022},
eprint = {2006.15726},
archivePrefix = {arXiv},
}
@misc{aim-helleseth-three-valued,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Excluding three correlation levels in quadratic field towers}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/616-helleseth-three-valued.md},
note = {Entry 616 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00352, "Excluding three correlation levels in quadratic field towers", clopen.solutions.