  We focus on the permutation polynomials of the form
$L(X)+\Tr_{m}^{3m}(X)^{s}$ over $\F_{q^3}$, where $\F_q$ is the finite field
with $q=p^m$ elements, $p$ is a prime number, $m$ is a positive integer,
$\Tr_{m}^{3m}$ is the relative trace function from $\F_{p^{3m}}$ to
$\F_{p^{m}}$, $L(X)$ is a linearized polynomial over $\F_{q^{3}}$, and $s>1$ is
a positive integer. More precisely, we present six new classes of permutation
polynomials over $\F_{q^3}$ of the aforementioned form: one class over finite
fields of even characteristic, three classes over finite fields of odd
characteristic, and the remaining two over finite fields of arbitrary
characteristic. Furthermore, we show that these classes of permutation
polynomials are inequivalent to the known ones of the same form. We also
provide the explicit expressions for the compositional inverses of each of
these classes of permutation polynomials.
