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

Let exp p ( q ) denote the number of times the prime number p appears in the prime factorization of the integer q . The following result is proved: If there is a perfect 1-error correcting code of length  n  over an alphabet with  q  symbols then, for every prime number  p , exp p ( 1 + n ( q − 1 ) ) ≤ exp p ( q ) ( 1 + ( n − 1 ) / q ) . ondition 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 ) .