  This paper presents a new achievable scheme for coded caching systems with
$\mathsf{N}$ files, $\mathsf{K}=\mathsf{N}$ users, and cache size
$\mathsf{M}=1/(\mathsf{N}-1)$. The scheme employs linear coding during the
cache placement phase, and a three-stage transmissions designed to eliminate
interference in the delivery phase. The achievable load meets a known converse
bound, which impose no constraint on the cache placement, and is thus optimal.
This new result, together with known inner and outer bounds, shows optimality
of linear coding placement for $\mathsf{M} \leq 1/(\mathsf{N}-1)$ when
$\mathsf{K}=\mathsf{N}\geq 3$. Interestingly and surprisingly, the proposed
scheme is relatively simple but requires operations on a finite field of size
at least 3.
