A method of construction of differentially $4$-uniform permutations over $V_{m}$ for even $m$

A generalization of the method of C. Carlet for constructing differentially 4-uniform permutations of binary vector spaces in even dimension $2k$ is suggested. It consists in restricting APN-functions in $2k+1$ variables to a linear manifold of dimension $2k$. The general construction of the method is proposed and a criterion for its applicability is established. Power permutations to which this construction is applicable are completely described and a class of suitable not one-to-one functions is presented.