AbstractIn this paper, we construct several new permutation polynomials over finite fields. First, using the linearised polynomials, we construct the permutation polynomial of the form
${ \mathop{\sum }\nolimits}_{i= 1}^{k} ({L}_{i} (x)+ {\gamma }_{i} ){h}_{i} (B(x))$
over
${\mathbf{F} }_{{q}^{m} } $
, where
${L}_{i} (x)$
and
$B(x)$
are linearised polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms
$xh({\lambda }_{j} (x))$
and
$xh({\mu }_{j} (x))$
, where
${\lambda }_{j} (x)$
is the
$j$
th elementary symmetric polynomial of
$x, {x}^{q} , \ldots , {x}^{{q}^{m- 1} } $
and
${\mu }_{j} (x)= {\mathrm{Tr} }_{{\mathbf{F} }_{{q}^{m} } / {\mathbf{F} }_{q} } ({x}^{j} )$
. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form
${L}_{1} (x)+ {L}_{2} (\gamma )h(f(x))$
over
${\mathbf{F} }_{{q}^{m} } $
, which extends a result of Kyureghyan.