  It was proved by Nill that for any lattice simplex of dimension $d$ with
degree $s$ which is not a lattice pyramid, the inequality $d+1 \leq 4s-1$
holds. In this paper, we give a complete characterization of lattice simplices
satisfying the equality, i.e., the lattice simplices of dimension $(4s-2)$ with
degree $s$ which are not lattice pyramids. It turns out that such simplices
arise from binary simplex codes. As an application of this characterization, we
show that such simplices are counterexamples for the conjecture known as
"Cayley conjecture". Moreover, by modifying Nill's inequaitly slightly, we also
see the sharper bound $d+1 \leq f(2s)$, where $f(M)=\sum_{n=0}^{\lfloor \log_2
M \rfloor} \lfloor M/2^n \rfloor$ for $M \in \mathbb{Z}_{\geq 0}$. We also
observe that any lattice simplex attaining this sharper bound always comes from
a binary code.
