Four Special Directions over Prime Fields: Structure and Complete Classification
Abstract
A direction of the affine plane 𝔽p2 is special for a set if the lines in that direction do not all meet the set in the same number of points. We classify, for every prime p≥5, the subsets of 𝔽p2 with exactly four special directions. There are $54$ affine-equivalence classes, occurring only for p=5,7,11,13,17, and we give explicit representatives. In particular, $17$ is the largest prime admitting such a set, which answers a question of Ghidelli, Kiss and Somlai, and over 𝔽13 the minimum size is $65$, attained by exactly two classes. The key structural fact is that the projection polynomials of such a set share a vertex: the set is either centrally symmetric about this point or differs from its reflection by exchanging one full line with one empty line. We also show that, over an odd prime field, a Boolean function on 𝔽pd with exactly four active projective Fourier lines is pulled back from a plane, so the classification extends to every dimension. The classification and the exclusion of the primes 19≤ p<2048 are computer-assisted; the structural results and the exclusion of larger primes are proved directly.
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BibTeX
@misc{wang2026four,
title = {{Four Special Directions over Prime Fields: Structure and Complete Classification}},
author = {Qihang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20495},
eprint = {2609.20495},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20495},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Four Special Directions over Prime Fields: Structure and Complete Classification
AU - Qihang Wang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.20495
UR - https://arxiv.org/abs/2609.20495
AB - A direction of the affine plane 𝔽_p^2 is special for a set if the lines in that direction do not all meet the set in the same number of points. We classify, for every prime p≥5, the subsets of 𝔽_p^2 with exactly four special directions. There are $54$ affine-equivalence classes, occurring only for p=5,7,11,13,17, and we give explicit representatives. In particular, $17$ is the largest prime admitting such a set, which answers a question of Ghidelli, Kiss and Somlai, and over 𝔽_13 the minimum size is $65$, attained by exactly two classes. The key structural fact is that the projection polynomials of such a set share a vertex: the set is either centrally symmetric about this point or differs from its reflection by exchanging one full line with one empty line. We also show that, over an odd prime field, a Boolean function on 𝔽_p^d with exactly four active projective Fourier lines is pulled back from a plane, so the classification extends to every dimension. The classification and the exclusion of the primes 19≤ p<2048 are computer-assisted; the structural results and the exclusion of larger primes are proved directly.
ER -