发表机构
Nanjing University of Posts and Telecommunications; Institute of Information Engineering, Chinese Academy of Sciences; Hefei University of Technology; Konkuk University（南京邮电大学; 中国科学院信息工程研究所; 合肥工业大学; 建国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了无限族Maiorana-McFarland弯曲函数，通过三角集合划分置换和坐标变换，证明其平移设计与经典辛设计同构，从而完全解决了Polujan-Pott猜想。
AI 中文摘要
本文通过构造一个显式的、无限族的Maiorana-McFarland弯曲函数$f_t$，解决了Polujan和Pott的一个猜想。该函数定义在$2(2^t-1)$个变量上，对于任意整数$t \ge 2$，其代数次数$°(f_t) = t + 1$。我们的构造基于一个极小的交换代数$I_t$，它自然地诱导出一个三角集合划分多项式置换$P_t$。通过在直和$I_t \oplus I_t^*$中识别出一个初等阿贝尔子群，我们建立了一个显式的非线性坐标变换，将$f_t$拉回到一个典型二次型。这在一个特殊群结构下将平移发展$\operatorname{Dev}(D_{f_t})$线性化，并证明它与经典辛设计$S^\pm(2(2^t-1))$同构，从而完全解决了该猜想。
英文摘要
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.
Comments16 pages
