Title: On the q-analogue of codes, matroids, simplicial complexes and their relations
Abstract: For a code with the Hamming metric one can associate a matroid using its parity check matrix. The independent elements of this matroid form a simplicial complex. Simplicial complexes associated to a q-matroid were shown to be shellable and this property helps to relate some properties of the simplicial complex to the codes. Indeed it was shown that there is a relation between Betti Numbers associated to the simplicial complex and the generalized weights of the linear code.
In the first part of this talk, we will give a brief review of the relations between these objects. A very interesting question is to find whether similar relations exist when we replace the Hamming metric with the rank metric. As a partial answer to this question, a q-analogue of some of the previous results are presented in the second part of the talk.
