Construction of partitions of the set of all $p$-ary vectors of length $p+1$ into Hamming codes

We suggest the construction of a partition of the set of all $p$‑ary vectors of length $p+1$ into perfect $p$-ary codes, where $p$ is a prime. The construction yields the lower bound $N(p)>(e^{\pi\sqrt{2p/3}})/(4p\sqrt{3})$ on the number of nonequivalent such partitions for any prime $p$.