  The Johnson graph $J(v, k)$ has as vertices the $k$-subsets of
$\mathcal{V}=\{1,\ldots, v\}$, and two vertices are joined by an edge if their
intersection has size $k-1$. An \emph{$X$-strongly incidence-transitive code}
in $J (v, k)$ is a proper vertex subset $\Gamma$ such that the subgroup $X$ of
graph automorphisms leaving $\Gamma$ invariant is transitive on the set
$\Gamma$ of `codewords', and for each codeword $\Delta$, the setwise stabiliser
$X_\Delta$ is transitive on $\Delta \times (\mathcal{V}\setminus \Delta)$. We
classify the \emph{$X$-strongly incidence-transitive codes} in $J(v,k)$ for
which $X$ is the symplectic group $\mathrm{Sp}_{2n}(2)$ acting as a
$2$-transitive permutation group of degree $2^{2n-1}\pm 2^{n-1}$, where the
stabiliser $X_\Delta$ of a codeword $\Delta$ is contained in a \emph{geometric}
maximal subgroup of $X$. In particular, we construct two new infinite families
of strongly incidence-transitive codes associated with the reducible maximal
subgroups of $\mathrm{Sp}_{2n}(2)$.
