Characterizations and MacWilliams-Like Identities of Weakly Regular Bent Partitions

Abstract

Bent partitions play a significant role in the theory of bent functions. In this paper, we study weakly regular bent partitions, that is, all generated bent functions are weakly regular (called in class 𝒲ℛℬ𝒫). Let Γ={A_i, i ∈ V_m} be a bent partition of V_n in class 𝒲ℛℬ𝒫, where V_n denotes an n-dimensional vector space over the prime field 𝔽_p, n is even, m ≤ n/2, and m ≥ 2 if p=2. We first characterize bent partition Γ in terms of the duals of component functions of vectorial bent functions. In particular, we prove that if p ≠ 3, then all bent functions generated by Γ are regular. Building on the above result, we give characterizations of bent partition Γ in terms of two classes of linear codes. When n ≥ 4 and m=n/2, except for the case where (p, n)=(2, 4) and |A_i₀|=1 for some i₀, we show that Γ is obtained from a spread (after a shift). Particularly, any normal bent partition of V_n of depth p^n/2 in class 𝒲ℛℬ𝒫 must be obtained from a spread (after a shift), where n ≥ 4. We establish the Macwilliams-like identities for bent partitions in class 𝒲ℛℬ𝒫. Furthermore, if p is odd, we present some nonexistence results on bent partitions in class 𝒲ℛℬ𝒫.

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BibTeX RIS

BibTeX

@misc{wang2026characterizations,
  title = {{Characterizations and MacWilliams-Like Identities of Weakly Regular Bent Partitions}},
  author = {Jiaxin Wang and Yadi Wei and Fang-Wei Fu and Jin Li and Fulin Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41779},
  eprint = {2610.41779},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41779},
}

RIS

TY  - UNPB
TI  - Characterizations and MacWilliams-Like Identities of Weakly Regular Bent Partitions
AU  - Jiaxin Wang
AU  - Yadi Wei
AU  - Fang-Wei Fu
AU  - Jin Li
AU  - Fulin Li
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41779
UR  - https://arxiv.org/abs/2610.41779
AB  - Bent partitions play a significant role in the theory of bent functions. In this paper, we study weakly regular bent partitions, that is, all generated bent functions are weakly regular (called in class 𝒲ℛℬ𝒫). Let Γ={A_i, i ∈ V_m} be a bent partition of V_n in class 𝒲ℛℬ𝒫, where V_n denotes an n-dimensional vector space over the prime field 𝔽_p, n is even, m ≤ n/2, and m ≥ 2 if p=2. We first characterize bent partition Γ in terms of the duals of component functions of vectorial bent functions. In particular, we prove that if p ≠ 3, then all bent functions generated by Γ are regular. Building on the above result, we give characterizations of bent partition Γ in terms of two classes of linear codes. When n ≥ 4 and m=n/2, except for the case where (p, n)=(2, 4) and |A_i_0|=1 for some i_0, we show that Γ is obtained from a spread (after a shift). Particularly, any normal bent partition of V_n of depth p^n/2 in class 𝒲ℛℬ𝒫 must be obtained from a spread (after a shift), where n ≥ 4. We establish the Macwilliams-like identities for bent partitions in class 𝒲ℛℬ𝒫. Furthermore, if p is odd, we present some nonexistence results on bent partitions in class 𝒲ℛℬ𝒫.
ER  -

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