Code Enumerators and Tutte Polynomials

• Published in: IEEE transactions on information theory Vol. 56; no. 9; pp. 4350 - 4358
• Main Author: Britz, Thomas
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Australia; Codes; Codeword m -tuple; Coding theory; Coding, codes; Exact sciences and technology; Galois fields; Hamming weight; higher weight enumerator; Information theory; Information, signal and communications theory; Linear code; MacWilliams identity; Mathematics; matroid; Polynomials; Signal and communications theory; Statistics; Telecommunications and information theory; tutte polynomial; Linear code; Finite field; higher weight enumerator; Matroid; MacWilliams identity; Codeword m-tuple; tutte polynomial
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2010.2054654

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.