The Last Seven Open Radii for Perfect Codes in the Johnson Scheme

Delsarte (1973) conjectured that there are no nontrivial perfect codes in the Johnson scheme. In this paper, we prove that there are no nontrivial $e$-perfect codes in the Johnson scheme for $e\in \{1,2,4,9,10,12,16\}$. This paper complements “Perfect Codes in the Johnson Scheme Hardly Exist”, thus proving Delsarte’s conjecture completely.