  Nitrides, carbides and borides of transition metals are an attractive class
of hard materials. Our recent preliminary explorations of the binary chemical
compounds indicated that chromium-based materials are among the hardest
transition metal compounds. Motivated by this, here we explore in detail the
binary Cr-B, Cr-C and Cr-N systems using global optimization techniques.
Calculated enthalpy of formation and hardness of predicted materials were used
for Pareto optimization to define the hardest materials with lowest energy. Our
calculations recover all numerous known stable compounds (except $Cr_{23}C_6$
with its large unit cell) and discover a novel stable phase $Pmn2_1$-$Cr_2C$.
We resolve the structure of $Cr_2N$ and find it to be of anti-$CaCl_2$ type
(space group Pnnm). Many of these phases possess remarkable hardness, but only
$CrB_4$ is superhard (Vickers hardness 48 GPa). Among chromium compounds,
borides generally possess highest hardnesses and greatest stability. Under
pressure, we predict stabilization of a TMDC-like phase of $Cr_2N$, a WC-type
phase of CrN, and a new compound $CrN_4$. Nitrogen-rich chromium nitride
$CrN_4$ is a high energy-density material featuring polymeric nitrogen chains.
In the presence of metal atoms polymerization of nitrogen takes place at much
lower pressures: $CrN_4$ becomes stable at 15 GPa.
