On the non-existence of extended perfect codes and some perfect colorings

Abstract

In this paper we obtain the necessary condition for the existence of perfect kcolorings (equitable k-partitions) in Hamming graphs H(n, q), where q = 2, 3, 4 and Doob graphs D(m, n). As an application, we prove the non-existence of extended perfect codes in H(n, q), where q = 3, 4, n > q + 2, and in D(m, n), where 2m + n > 6.