On the Nonexistence of Perfect Codes in $J(2w+p^2,w)$

Abstract

We consider the nonexistence of $e$-perfect codes in the Johnson scheme $J(n,w)$. It is proved that for each $J(2w+3p,w)$ for $p$ prime and $p\ne 2,5$, $J(2w+5p,w)$ for $p$ prime and $p\ne 3$, and $J(2w+p^2,w)$ for $p$ prime, it does not contain non-trivial $e$-perfect codes.