Recursively-generated permutations of a binary space

Nonlinear permutations of a vector space $GF(2)^m$ of any dimension $m\ne2^t$, $t\in\mathbb N$, induced by iterations of linear transformation over the ring $R=\mathbb Z_4$ with characteristic polynomial $F(x)\in R[x]$, $F(x)\equiv(x\oplus e)^m\pmod2$, are studied.