A Novel Approach to Counterexamples of the Polujan-Pott Conjecture via Set-Partition Permutations
Abstract
In this paper, we settle a conjecture of Polujan and Pott by constructing an explicit, infinite family of Maiorana--McFarland bent functions ft in 2(2t−1) variables with algebraic degree °(ft) = t + 1 for any integer t ≥ 2. Our construction builds upon a minimal commutative algebra It, which naturally induces a triangular set-partition polynomial permutation Pt. By identifying an elementary abelian subgroup within the direct sum $ I_t \oplus I_t^*$, we establish an explicit nonlinear coordinate transformation that pulls ft back to a canonical quadratic form. This linearizes the translation development Dev(Dft) under an exotic group structure and proves that it is isomorphic to the classical symplectic design S±(2(2t−1)), thereby fully resolving the conjecture.
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BibTeX
@misc{wu2026novel,
title = {{A Novel Approach to Counterexamples of the Polujan-Pott Conjecture via Set-Partition Permutations}},
author = {Yansheng Wu and Jiaxin Wang and Jong Yoon Hyun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13968},
eprint = {2609.13968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13968},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Novel Approach to Counterexamples of the Polujan-Pott Conjecture via Set-Partition Permutations
AU - Yansheng Wu
AU - Jiaxin Wang
AU - Jong Yoon Hyun
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13968
UR - https://arxiv.org/abs/2609.13968
AB - In this paper, we settle a conjecture of Polujan and Pott by constructing an explicit, infinite family of Maiorana--McFarland bent functions f_t in 2(2^t−1) variables with algebraic degree °(f_t) = t + 1 for any integer t ≥ 2. Our construction builds upon a minimal commutative algebra I_t, which naturally induces a triangular set-partition polynomial permutation P_t. By identifying an elementary abelian subgroup within the direct sum $ I_t \oplus I_t^*$, we establish an explicit nonlinear coordinate transformation that pulls f_t back to a canonical quadratic form. This linearizes the translation development Dev(D_f_t) under an exotic group structure and proves that it is isomorphic to the classical symplectic design S^±(2(2^t−1)), thereby fully resolving the conjecture.
ER -