Nonexistence of Perfect Codes for Some Composite Alphabets

It is known [V. A. Zinoviev and V. K. Leont'ev, Probl. Control Inf. Theory, 1973, vol. 2, no. 2, pp. 123–132; A. Tietäväinen, SIAM J. Appl. Math., 1973, vol. 24, no. 1, pp. 88–96] that if the cardinality of an alphabet $q$ is a power of a prime number, then nontrivial perfect codes other than the Hamming and Golay codes do not exist. A natural assumption is that this is true for composite $q$. In this paper it is shown that there do not exist nontrivial perfect codes over an alphabet of $q =2^{\alpha}3^{\beta}$ ($\alpha$,$\beta\geq l)$ symbols that correct $t\geq 2$ errors. The question remains an open one for $t=1$. The only known result in this case [S. W. Golomb and E. S. Posner, IEEE Trans. Inf. Theory, 1964, vol. 10, no. 1, pp. 196–208] is that a perfect single-error-correcting code does not exist for $q=6$ and $n=7$.