MacWilliams Identities and Matroid Polynomials
نویسنده
چکیده
We present generalisations of several MacWilliams type identities, including those by Kløve and Shiromoto, and of the theorems of Greene and Barg that describe how the Tutte polynomial of the vector matroid of a linear code determines the rth support weight enumerators of the code. One of our main tools is a generalisation of a decomposition theorem due to Brylawski.
منابع مشابه
Latroids and Their Representation by Codes over Modules
It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code i...
متن کاملMacWilliams 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. W...
متن کاملMacwilliams Duality and the Rosenbloom–tsfasman Metric
A new non-Hamming metric on linear spaces over finite fields has recently been introduced by Rosenbloom and Tsfasman [8]. We consider orbits of linear groups preserving the metric and show that weight enumerators suitably associated with such orbits satisfy MacWilliams-type identities for mutually dual codes. Furthermore, we show that the corresponding weight spectra of dual codes are related b...
متن کاملConvolution-multiplication identities for Tutte polynomials of matroids
Abstract. We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specialization of this identity. Combinatorial or probabilistic interpr...
متن کاملWEIGHT POLYNOMIALS OF SELF-DUAL CODES AND THE MacWILLIAMS IDENTITIES
Many error correcting codes are known to be self-dual. Hence the MacWilliams identities put a considerable restriction on the possible weight distribution of such a code. We show that this restriction, for codes over GF(2) and GF(3), is that the weight polynomial must lie in an explicitly described free polynomial ring. To extend these results (in part) to self-dual codes over larger fields, we...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Electr. J. Comb.
دوره 9 شماره
صفحات -
تاریخ انتشار 2002