The non-existence of some perfect codes over non-prime power alphabets

Let expp(q) denote the number of times the prime number pp appears in the prime factorization of the integer qq. The following result is proved: If there is a perfect 1-error correcting code of length   nn over an alphabet with   qq symbols then, for every prime number   p,expp(1+n(q−1))≤expp(q)(1+(n−1)/q).This condition is stronger than both the packing condition and the necessary condition given by the Lloyd theorem, as it for example excludes the existence of a perfect code with the parameters (n,q,e)=(19,6,1)(n,q,e)=(19,6,1).