MacWilliams identities and matroid polynomials
In 1963, MacWilliams proved an identity linking the weight enumerators of a linear code and its dual. In 1976, Greene showed that the weight enumerator of a linear code could be expressed in terms of the Tutte polynomial of the matroid that is represented by a generator matrix of the code. Using this fact, Greene was able to reprove the identity of MacWilliams. This paper proves several generalizations of the MacWilliams identity including results that extend earlier generalizations due to Kløve and Shiromoto. In addition, the theorem of Greene and an extension of it due to Barg are also generalized. An important tool in these last two proofs is an extension of Brylawski's characterization of the Tutte polynomial.
scientific article; zbMATH DE number 1741005