In 1970 on the ICM in Nice, A.M. Gleason presented his famous theorem that the weight enumerator of a doubly-even self-dual binary code is an element of the polynomial ring generated by the weight enumerators of the Hamming code of length 8 and the Golay code of length 24. The proof uses the fact that this polynomial ring is the invariant ring of the complex reflection group G9 of order 192.

In the meantime, many variations of this theorem have been proven. Together with E. Rains and N. Sloane, we develop a theory that allows us to prove that, in a quite general situation, the weight enumerators of codes of a given Type over a not necessary commutative finite ring span the invariant ring of the associated Clifford-Weil group. These are finite complex matrix groups given by explicit generators.