A Novel Approach to Counterexamples of the Polujan-Pott Conjecture via Set-Partition Permutations

2026-09-12

AI Summary

本文通过构建特定的代数结构和非线性坐标变换，解决了Polujan-Pott猜想，提供了一种生成反例的新方法。

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.

Problem

• Polujan-Pott Conjecture
• Maiorana-McFarland bent functions
• algebraic degree

Innovation

• Maiorana-McFarland bent functions
• set-partition permutations
• algebraic degree
• nonlinear coordinate transformation
• symplectic design