On the dependency between the code symmetries and the decoding efficiency

A framework of monomial codes is considered, which includes linear codes generated by the evaluation of certain monomials. Polar and Reed-Muller codes are the two best-known representatives of such codes and can be considered as two extreme cases. Reed-Muller codes have a large automorphism group but their low-complexity maximum likelihood decoding still remains an open problem. On the other hand, polar codes have much less symmetries but admit the efficient near-ML decoding.