In the seminal paper Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math 89 (1999) 227-262, E. Lindenstrauss initiated the study of the relation between embeddability and mean dimension and raised the corresponding question for (minimal) $Z^k$-actions. We solve the problem by showing that minimal systems of mean dimension less than $N/2$ embed equivariantly into the cubical shift $([0,1]^N)^{Z^k}$. The result is optimal and improves upon a previous result by Gutman, Lindenstrauss and Tsukamoto. The proof uses multi-dimensional sampling theory and is significantly harder than the proof of the case $k=1$ which was settled previously by Gutman and Tsukamoto. Joint work with Yixiao Qiao and Masaki Tsukamoto.