Perfect codes of complete rank with kernels of large dimensions

We construct perfect codes of all admissible lengths $n>20^{10}-1$ of complete rank with kernels of all possible dimensions $K$ from $(n-1)/2$ to $U(n)$, which is the maximum possible. For every $k\in \{(n-1)/2,\dots,U(n)-2\}$, we construct such codes of length $n,31\leqslant n\leqslant 2^{10}-1$.