Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: MacWilliams identities and matroid polynomials. | MacWilliams identities and matroid polynomials Thomas Britz Department of Mathematical Sciences University of Aarhus Denmark britz@ Submitted December 12 2001 Accepted April 22 2002 MR Subject Classifications 05B35 94B05 Abstract We present generalisations of several MacWilliams type identities including those by Klrtve 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. 1 Introduction Since the 1963 article 8 by F. J. MacWilliams coding theorists have paid considerable attention to the support Hamming weight distribution of linear codes. In later years this interest has increased due to results such as those by Wei 16 on rth generalised Hamming weights Klpve 6 and Simonis 13 on rth support Hamming weight distributions effective length distributions in Simonis terminology and Shiromoto 11 on A-ply weight enumerators. Section 2 of this paper introduces notation and the various enumerators by presenting the MacWilliams identities 8 as well as their generalisations by Klpve 6 and Shiromoto 11 . The two main results of this section Theorems 3 and 7 generalise these results. Proofs of these theorems appear in the later sections. In Section 3 we generalise theorems due to Greene 5 and Barg 1 that describe how the Tutte polynomial of the vector matroid of a linear code determines the rth support weight enumerators of the code. We obtain two theorems which turn out to be equivalent to each other and to the Critical Theorem by Crapo and Rota 4 . The main tool is a generalisation of the characterisation of Tutte-Groethendieck polynomials due to Brylawski 3 . As applications of these theorems we prove Theorems 3 and 7 of Section 2. In Section 4 an alternative proof of Theorem 7 is presented. This proof relies on coding-theoretical arguments .
