Hamming cube as an L1 and L2 metric ($H_n=\{0,1{\}}^n$)

The Hamming cube has $d_H(x,y)={\sum }_j|x_j-y_j|$. Its coordinate map into ${\ell }_1^n$ is isometric, while the same map into ${\ell }_2^n$ has distance $\sqrt{d_H(x,y)}$. Thus the family of all Hamming cubes uniformly coarsely embeds into both $L^1$ and $L^2$.