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, which can be considered as two extreme cases from a certain point of view... We introduce a new family of codes, partially symmetric codes. Partially symmetric monomial codes have a smaller group of automorphisms than RM codes and are in some sense "between" Reed-Muller and polar codes. A lower bound on their parameters is introduced along with the explicit construction which achieves it. Structural properties of these codes are demonstrated and it is shown that in some cases partially monomial codes also have a recursive structure. (read more)