Lectures on Codes, matroids, their links & q-analogues I

2017年   1月 31日 14時 00分 ~ 2017年   1月 31日 16時 00分
room 4-403
Relinde Jurrius (Univ. Neuchatel, Switzerland)
Abstracts: The weight enumerator is an important invariant of a linear code. Its determination is the starting point of the first talk. We will show a method to calculate the weight enumerator that highlights the connection to the Tutte polynomial of the associate matroid. We will do the same thing for rank metric codes: these codes use the rank metric instead of the Hamming metric. This leads to the rank weight enumerator. We will see that rank metric codes are the q-analogue of linear codes. A q-analogue is, roughly speaking, what happens when we generalise from finite sets to finite spaces.

In the second talk, we will try to find a q-analogue of the link between the weight enumerator and the Tutte polynomial. For this, we need the q-analogue of a matroid. Unfortunately, equivalent definitions of a matroid do not need to have equivalent q-analogues! We will see different approaches to defining the q-analogue of a matroid and argue what we think is the best definition.

(In theory, the two talks can be followed independently. Some prior encounter with matroid theory is very helpful in understanding the talks.)