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Recently, there has been a lot of work on constructions of permutationpolynomials of the form $(x^{2^m}+x+\delta)^{s}+x$ over the finite field$\F_{2^{2m}}$, especially in the case when $s$ is of the form $s=i(2^m-1)+1$(Niho exponent). In this paper, we further investigate permutation polynomialswith this form. Instead of seeking for sporadic constructions of the parameter$i$, we give a general sufficient condition on $i$ such that$(x^{2^m}+x+\delta)^{i(2^m-1)+1}+x$ permutes $\F_{2^{2m}}$, that is, $(2^k+1)i\equiv 1 ~\textrm{or}~ 2^k~(\textrm{mod}~ 2^m+1)$, where $1 \leq k \leq m-1$ isany integer. This generalizes a recent result obtained by Gupta and Sharma whoactually dealt with the case $k=2$. It turns out that most of previousconstructions of the parameter $i$ are covered by our result, and it yieldsmany new classes of permutation polynomials as well.

A general construction of permutation polynomials of the form $ (x^{2^m}+x+\delta)^{i(2^m-1)+1}+x$ over $\F_{2^{2m}}$