On Perfect Codes and Related Concepts
Designs, Codes and Cryptography
Codes and anticodes in the Grassman graph
Journal of Combinatorial Theory Series A
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Although it was conjectured by Delsarte in 1973 that no nontrivial perfect codes exist in the Johnson scheme, only very partial results are known. In this paper we considerably reduce the range in which perfect codes in the Johnson scheme can exist; e.g., we show that there are no nontrivial perfect codes in the Johnson graph $J(2w+p,w)$, $p$ prime. We give theorems about the structure of perfect codes if they exist. This involved structure gives more evidence in support of the belief that no nontrivial perfect codes exist in the Johnson scheme.