О разбиениях пространства $F^N_q$ на аффинно неэквивалентные совершенные $q$-значные коды

It is proved that there exists a partition of the set $F^N_q$ of all $q$-ary vectors of length $N$ into pairwise affine nonequivalent perfect $q$-ary codes of length $N$ with the Hamming distance $3$ for any $N=(q^m-1)/(q-1)$, where $q=p^r,$ $p$ is prime.