Configuration Distribution and Designs of Codes in the Johnson Scheme

The main goal of this article is to present several connections between perfect codes in the Johnson scheme and designs, and provide new tools for proving Delsarte conjecture that there are no nontrivial perfect codes in the Johnson scheme. Three topics will be considered. The first is the configuration distribution which is akin to the weight distribution in the Hamming scheme. We prove that if there exists an e-perfect code C in the Johnson scheme then there is a formula which connects the number of vectors at distance i from any codeword in various codes isomorphic to C. The second topic is the Steiner systems embedded in a perfect code. We prove a lower bound on the number of Steiner systems embedded in a perfect code. The last topic is the strength of a perfect code. We show two new methods for computing the strength of a perfect code and demonstrate them on 1-perfect codes. We further discuss how to settle Delsarte conjecture. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 15–34, 2007