Multidimensional second order Reed-Muller codes as Grassmannian packings

We derive a generalization of a result in representation theory. Using this generalization, we construct new families of Grassmannian packings associated with binary Reed-Muller codes and we develop a low complexity decoding algorithm by modifying standard decoding algorithms for these binary codes. The subspaces are associated with projection operators which arise in the theory of quantum stabilizer codes. These Grassmannian packings find application as highly structured examples of dictionaries that admit fast algorithms for identifying sparse representations, and in noncoherent wireless communication with multiple antennas. The capacity of the noncoherent MIMO channel at both low and moderate SNR (under the constraint that only isotropically distributed unitary matrices are used for information transmission) is closely approximated by these packings.