Classifying polynomials of linear codes

Relinde Jurrius

The weight enumerator of a linear code is a classifying polynomial associated with the code. Besides its intrinsic importance as a mathematical object, it is used in the probability theory around codes. For example, the weight enumerator of a binary code is very useful if we want to study the probability that a received message is closer to a different codeword than to the codeword sent. (Or, rephrased: the probability that a maximum likelihood decoder makes a decoding error.)

We will generalize the weight enumerator in two ways, which lead to polynomials which are better invariants for a code. A procedure for the determination of these polynomials is given. We will show that the two generalisations determine each other, and that they connect to the Tutte polynomial of a matroid, thus linking coding theory and matroid theory. Most of the generalisations and connections have been studied before, but mostly only one-way, and a complete overview was never given. We will use the established connections to derive MacWilliams relations for our generalisations.

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