The following expository article is intended to describe a correspondence between matroids and codes. The key results are that the weight enumerator of a code is a specialisation of the Tutte polynomial of the corresponding matroid, and that the MacWilliams relation between weight enumerators of a code and its dual can be obtained from matroid duality. It also provides a general introduction to matroids, an introduction to trellis decoding, and an algebraic construction of the minimal trellis of a code. Some of this material was presented in the QMW study group, although this version is my own re-working of it. I am grateful to Carrie Rutherford and Costas Papadopoulos for their contributions. Some of Carrie’s are acknowledged in the text, while Costas taught me about trellis decoding.
[1]
James G. Oxley,et al.
Matroid theory
,
1992
.
[2]
Dominic Welsh,et al.
The Tutte polynomial
,
1999,
Random Struct. Algorithms.
[3]
V. Vazirani,et al.
The "art of trellis decoding" is computationally hard-for large fields
,
1998,
Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).
[4]
C. Greene.
Weight Enumeration and the Geometry of Linear Codes
,
1976
.
[5]
Douglas J. Muder.
Minimal trellises for block codes
,
1988,
IEEE Trans. Inf. Theory.
[6]
H. Crapo,et al.
The Tutte polynomial
,
1969,
1707.03459.