Propelinear 1-perfect codes from quadratic functions

Perfect codes obtained by the Vasil’ev–Sch"onheim construction from a linear base code and quadratic switching functions are transitive and, moreover, propelinear. This gives at least $\exp (c{N}^2)$ propelinear $1$-perfect codes of length $N$ over an arbitrary finite field, while an upper bound on the number of transitive codes is $\exp (C(N\ln N{)}^2)$. Keywords: perfect code, propelinear code, transitive code, automorphism group, Boolean function.