A Novel Approach to Counterexamples of the Polujan-Pott Conjecture via Set-Partition Permutations
Year: 2026
Type: preprint
Abstract: 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 $t \ge 2$. 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 $\operatorname{Dev}(D_{f_t})$ under an exotic group structure and proves that it is isomorphic to the classical symplectic design $S^\pm(2(2^t-1))$, thereby fully resolving the conjecture.
Source: arXiv (Cornell University)
Authors: Wu, Yansheng, Wang, Jiaxin, Hyun, Jong Yoon
Institution: Nanjing University of Posts and Telecommunications
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3862649344, G5792947667
OpenAlex ID: W7213350792