  In EUROCRYPT 2018, Cid et al. \cite{BCT2018} introduced a new concept on the
cryptographic property of S-boxes: Boomerang Connectivity Table (BCT for short)
for evaluating the subtleties of boomerang-style attacks. Very recently, BCT
and the boomerang uniformity, the maximum value in BCT, were further studied by
Boura and Canteaut \cite{BC2018}. Aiming at providing new insights, we show
some new results about BCT and the boomerang uniformity of permutations in
terms of theory and experiment in this paper. Firstly, we present an equivalent
technique to compute BCT and the boomerang uniformity, which seems to be much
simpler than the original definition from \cite{BCT2018}. Secondly, thanks to
Carlet's idea \cite{Carlet2018}, we give a characterization of functions $f$
from $\mathbb{F}_{2}^n$ to itself with boomerang uniformity $\delta_{f}$ by
means of the Walsh transform. Thirdly, by our method, we consider boomerang
uniformities of some specific permutations, mainly the ones with low
differential uniformity. Finally, we obtain another class of $4$-uniform BCT
permutation polynomials over $\mathbb{F}_{2^n}$, which is the first binomial.
