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 $\mathcal{WR}\mathcal{B}\mathcal{P}$). Let $\Gamma =\{A_i,i\in V_m\}$ be a bent partition of $V_n$ in class $\mathcal{WR}\mathcal{B}\mathcal{P}$, where $V_n$ denotes an $n$-dimensional vector space over the prime field ${\mathbb{F}}_p$, $n$ is even, $m\le \frac{n}{2}$, and $m\ge 2$ if $p=2$. We first characterize bent partition $\Gamma$ in terms of the duals of component functions of vectorial bent functions. In particular, we prove that if $p\ne 3$, then all bent functions generated by $\Gamma$ are regular. Building on the above result, we give characterizations of bent partition $\Gamma$ in terms of two classes of linear codes. When $n\ge 4$ and $m=\frac{n}{2}$, except for the case where $(p,n)=(2,4)$ and $|A_{i_0}|=1$ for some $i_0$, we show that $\Gamma$ is obtained from a spread (after a shift). Particularly, any normal bent partition of $V_n$ of depth $p^{\frac{n}{2}}$ in class $\mathcal{WR}\mathcal{B}\mathcal{P}$ must be obtained from a spread (after a shift), where $n\ge 4$. We establish the Macwilliams-like identities for bent partitions in class $\mathcal{WR}\mathcal{B}\mathcal{P}$. Furthermore, if $p$ is odd, we present some nonexistence results on bent partitions in class $\mathcal{WR}\mathcal{B}\mathcal{P}$.