Complete Walsh spectra for a permutation-inverse family of Boolean functions

Let $q=2^e$ with $e$ even, and let ${\mathbb{F}}_{q^2}$ be the finite field of order $q^2$. Put $d=(q^2+q+1)\mathrm{/}3$, and consider the permutation polynomial $$\sigma (X)=X+X^d+X^{dq}\in {\mathbb{F}}_{q^2}[X]\mathrm{.}$$ For $\alpha \in {\mathbb{F}}_q^{\ast }$, define the Boolean function $$f_{\alpha }(x)={\text{Tr}}_{q^2}(\alpha ({\sigma }^{-1}(x){)}^3),x\in {\mathbb{F}}_{q^2},$$ where ${\text{Tr}}_{q^2}$ denotes the absolute trace from ${\mathbb{F}}_{q^2}$ to ${\mathbb{F}}_2$. In this paper, we determine all Walsh values of $f_{\alpha }$ and their multiplicities. In particular, $f_{\alpha }$ is bent if and only if $\alpha$ is a noncube in ${\mathbb{F}}_q$, proving a conjecture of Li, Li, Helleseth, and Qu. The $\beta \in {\mathbb{F}}_q$ part of the spectrum is handled by an elementary finite-field argument, whereas the $\beta \in {\mathbb{F}}_{q^2}\setminus {\mathbb{F}}_q$ part is reduced to a Hadamard problem on the trace-zero space. We then identify the resulting word with a normalized odd-dimensional shortened Kerdock shell in the noncube case and with a normalized first shortened Delsarte–Goethals shell in the cube case. To make the external shell input independently checkable, we separate the finite-field dictionary from the cited shell theorem and verify directly the degenerate case $e=2$. As an application, we show that a recent cyclotomic family of Xie, Li, Wang, and Zeng is the same construction in different coordinates.