# The Matroid of Supports of A Linear Code

@article{Barg1997TheMO, title={The Matroid of Supports of A Linear Code}, author={Alexander M. Barg}, journal={Applicable Algebra in Engineering, Communication and Computing}, year={1997}, volume={8}, pages={165-172} }

Abstract. A relation between the Hamming weight enumerator of a linear code and the Tutte polynomial of the corresponding matroid has been known since long ago. It provides a simple proof of the MacWilliams equation (see D. Welsh, Matroid Theory (1976)). In this paper we prove analogous results for the support weight distributions of a code.

## 47 Citations

### ON SOME POLYNOMIALS RELATED TO WEIGHT ENUMERATORS OF LINEAR CODES∗

- Mathematics, Computer Science
- 2002

Borders on all-terminal reliability of linear matroids are obtained and new proofs of two known results are obtained: Greene’s theorem and a connection between the weight polynomial and the partitions of the Potts model.

### Code Enumerators and Tutte Polynomials

- Computer ScienceIEEE Transactions on Information Theory
- 2010

A very general MacWilliams-type identity for linear codes that generalizes most previous extensions of the MacWilliams identity is proved and is applied to codeword m-tuples.

### Matroids and Codes with the Rank Metric

- Mathematics, Computer Science
- 2018

A Greene type identity is proved for the rank generating function of these matroidal structures and the rank weight enumerator of these linear codes and a combinatorial proof of a MacWilliams type identity for Delsarte rank-metric codes is given.

### MacWilliams Identities and Matroid Polynomials

- MathematicsElectron. J. Comb.
- 2002

Generalisations of several MacWilliams type identities and of the theorems of Greene and Barg that describe how the Tutte polynomial of the vector matroid of a linear code determines the support weight enumerators of the code are presented.

### Codes with the rank metric and matroids

- Computer Science, MathematicsDesigns, Codes and Cryptography
- 2018

A Greene type identity is proved for the rank generating function of these matroidal structures and the rank weight enumerator of these linear codes and a combinatorial proof of the MacWilliams type identity for Delsarte rank-metric codes is given.

### Codes , arrangements and weight enumerators

- Mathematics
- 2009

We treat error-correcting codes and several measures of error probability. In particular the weight enumerator and its generalization and extension will be considered from several points of view: 1)…

### On g-th MDS Codes and Matroids

- Mathematics, Computer ScienceAAECC
- 2003

This paper gives a relationship between the generalized Hamming weights for linear codes over finite fields and the rank functions of matroids and determines the support weight distributions of g-th MDS codes.

### MacWilliams Identities for m-tuple Weight Enumerators

- Computer Science, MathematicsSIAM J. Discret. Math.
- 2014

A generalization of theorems of Britz and of Ray-Chaudhuri and Siap is proved, which build on earlier work of Klo ve, Shiromoto, Wan, and others.

### Extended and Generalized Weight Enumerators

- Mathematics
- 2009

This paper gives a survey on extended and generalized weight enumerators of a linear code and the Tutte polynomial of the matroid of the code (16). Furthermore ongoing research is reported on the…

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