The MacWilliams relations are one of the gems or Coding Theory. They relate the weight enumeartor of a code and its orthogonal. They were first proved for binary codes by the great Jessie MacWilliams.
I have written a couple of unpublished short notes giving proofs of the MacWilliams
relations for certain weight enumerators for some very general rings.
You can read them:
Return to my home page.