MacWilliams Identities for $m$-tuple Weight Enumerators

Since MacWilliams proved the original identity relating the Hamming weight enumerator of a linear code to the weight enumerator of its dual code there have been many different generalizations, leading to the development of m-tuple support enumerators. We prove a generalization of theorems of Britz and of Ray-Chaudhuri and Siap, which build on earlier work of Kløve, Shiromoto, Wan, and others. We then give illustrations of these m-tuple weight enumerators. In a 1963 article [9], MacWilliams gave an identity relating the weight enumerator of a linear code to the weight enumerator of its dual code. Several authors have generalized this work in a few different directions. One type of generalization leads to weight enumerators in more than two variables, such as the Lee and complete weight enumerators, and to weight enumerators for codes defined over alphabets other than Fq. For example, a MacWilliams theorem for codes over Galois rings was given by Wan [17]. Another type of generalization considered by several authors is to adapt the notion of weight to consider more than one codeword at a time. This leads to the generalized Hamming weights of Wei [18], and to the MacWilliams type results for m-tuple support enumerators of Kløve [8], Shiromoto [14], Simonis [16], and Ray-Chaudhuri and Siap [12, 13]. Barg [1], and later Britz [2, 3], generalized some of these results and gave matroid-theoretic proofs. Britz [4] also recently described new and broad connections between weight enumerators and Tutte polynomials of matroids. We prove a MacWilliams type result that implies the two main theorems of Britz [2], which concern support weight enumerators of codes and in turn imply the earlier results of Kløve [8], Shiromoto [14], and Barg [1]. Our result also implies the main theorems of Ray-Chaudhuri and Siap [12, 13] giving MacWilliams theorems for complete weight enumerators of an m-tuple of codes C1, C2, . . . , Cm that are not necessarily the same. As in [13], we phrase our results in terms of codes over Galois rings instead of restricting ourselves to codes over fields. One key feature of our result is that not only can the codes C1, C2, . . . , Cm be distinct, but they do not necessarily have to be defined over the same ring, a generalization suggested in Siap’s thesis [15]. This is not the first MacWilliams theorem for m-tuples of codes defined over different alphabets. In [3], Britz gives such a result for codes defined over finite fields that are not necessarily the same, but there is an additional constraint that the codes must have the same vector matroid. This result is phrased in terms of code structure families, a direction we will not pursue here. Date: January 14, 2014. 2010 Mathematics Subject Classification. 94B05.