Embedding in q-ary 1-perfect codes and partitions

It is proved that every 1-error-correcting code over a finite field can be embedded in a 1-perfect code of some larger length. Embedding in this context means that the original code is a subcode of the resulting 1-perfect code and can be obtained from it by repeated shortening. Further, the result is generalized to partitions: every partition of the Hamming space into 1-error-correcting codes can be embedded in a partition of a space of some larger dimension into 1-perfect codes. For the partitions, the embedding length is close to the theoretical bound for the general case and optimal for the binary case.