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It is proved that the set of higher weight enumerators of a linear code over a finite field is equivalent to the Tutte polynomial associated to the code. An explicit expression for the Tutte polynomial is given in terms of the subcode weights. Generalizations of these results are proved and are applied to codeword m-tuples. These general results are used to prove a very general MacWilliams-type identity for linear codes that generalizes most previous extensions of the MacWilliams identity. In addition, a general and very useful matrix framework for manipulating weight and support enumerators of linear codes is presented.