We present a study of a highly symmetric crystal made of exclusively sp2 bonded atoms. Calculations of the structural and electronic properties are performed within the pseudopotential-density-functional approach for two different compositions made of (i) pure carbon and (ii) carbon and nitrogen compound. In both solids, one of the carbon-carbon bond lengths is found to be 1.35A, which is considerably smaller than any carbon-carbon bond length found in other carbon solids. The bulk moduli are calculated to be 241 and 286 GPa for the pure carbon and the carbon-nitride compounds, respectively. We demonstrate that the relatively low bulk moduli, considering the short bond lengths found in the structure, is due to the disruption of the carbon π bonding states. This is probably unavoidable when trying to form a three-dimensional structure out of a planar configuration like the sp2 bonds. The calculated density of states and band structures show that the pure carbon form is metallic whereas the carbon nitride is semiconducting. When carbon atoms are added to the interstitial regions, the carbon solid becomes insulating and the bulk modulus increases to 282 GPa.