Fine Difference Structure and Prime-Power Depth of Bent Partitions

Abstract

A p-ary bent partition of 𝔽_pⁿ is a partition into K nonempty cells such that every balanced assignment of its cells to 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 pⁿ/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| pⁿ, so K=p^t; nonempty cells further give 1≤ t<n. In even dimension, the classical cell-size theorem yields K| p^n/2. Together with the known odd-dimensional ternary three-fibre parameter restriction, this gives the global bound t≤⌊ n/2⌋. 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.

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BibTeX RIS

BibTeX

@misc{wu2026fine,
  title = {{Fine Difference Structure and Prime-Power Depth of Bent Partitions}},
  author = {Zhaorui Wu},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28133},
  eprint = {2608.28133},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28133},
}

RIS

TY  - UNPB
TI  - Fine Difference Structure and Prime-Power Depth of Bent Partitions
AU  - Zhaorui Wu
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28133
UR  - https://arxiv.org/abs/2608.28133
AB  - A p-ary bent partition of 𝔽_p^n is a partition into K nonempty cells such that every balanced assignment of its cells to 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| p^n, so K=p^t; nonempty cells further give 1≤ t<n. In even dimension, the classical cell-size theorem yields K| p^n/2. Together with the known odd-dimensional ternary three-fibre parameter restriction, this gives the global bound t≤⌊ n/2⌋. 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.
ER  -

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