On Perfect Codes and Related Concepts
Designs, Codes and Cryptography
On new completely regular q-ary codes
Problems of Information Transmission
On the regularity of perfect 2-colorings of the Johnson graph
Problems of Information Transmission
IEEE Transactions on Information Theory
Perfect Single Error-Correcting Codes in the Johnson Scheme
IEEE Transactions on Information Theory
On weight distributions of perfect colorings and completely regular codes
Designs, Codes and Cryptography
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The problem of the existence of perfect 2-colorings in Johnson graphs J(6,3) and J(7,3) is solved in this paper. Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte [3]. This problem of existence of such structures is closely related to Delsarte hypothesis about the nonexistence of nontrivial perfect codes in Johnson graphs, the problem of existence of block schemes, the problem of existence of completely regular codes in Johnson graphs and other well-known mathematical problems. Some auxiliary theorems, which can be applied for treatment of perfect colorings in two colors in other graphs, are given in this paper.