\begin{abstract}
Cohn, Kumar and Minton constructed a four-parameter family of tight $39$-point
codes in the octonionic projective plane. We show that near each member outside
a closed nowhere-dense set of resonant parameters, the labeled tight codes
modulo $F_4$ form exactly four ten-dimensional families, any two meeting in the
four-parameter family, although the first-order deformations of the member form
a $28$-dimensional space modulo $F_4$. The originally constructed code is
resonant, and we do not determine the tight codes near it; but every
neighborhood of it in the space of unlabeled tight $39$-point codes modulo
$F_4$ contains a nonempty
open subset homeomorphic to an open subset of $\mathbb R^{10}$, so any
neighborhood of it that is a manifold has dimension ten.
\end{abstract}