Fine Difference Structure and Prime-Power Depth of Bent Partitions

2026-08-28

AI Summary

研究解决了p元弯曲分区的深度K是否为p的幂的问题，通过证明每个非零h下精细单元格的平移不变性，确认了K=p^t，并限定了t的范围。

Abstract

A $p$-ary bent partition of $\mathbb{F}_p^n$ is a partition into $K$ nonempty cells such that every balanced assignment of its cells to $\mathbb{F}_p$ produces a bent function. It was asked whether every possible depth $K$ is a power of $p$; for general $p$, previous affirmative results required regularity or cell-symmetry hypotheses. We prove the stronger unconditional statement that, for every nonzero $h$, exactly $p^n/K$ points remain in the same fine cell under translation by $h$. Thus the fine cells form a partitioned difference family and the fine label map is zero-difference balanced. Consequently $K\mid p^n$, so $K=p^t$; nonempty cells further give $1\le t<n$. In even dimension, the classical cell-size theorem yields $K\mid p^{n/2}$. Together with the known odd-dimensional ternary three-fibre parameter restriction, this gives the global bound $t\le\lfloor n/2\rfloor$. The proof is an exact finite average over balanced coarsenings. The main counting identity and selected consequences are formalized and kernel-checked in Lean 4.

Problem

• bent partition
• prime-power depth
• difference family
• zero-difference balanced

Innovation

• partitioned difference family
• zero-difference balanced
• formal verification
• Lean 4