Codes, matroids and derived matroids

Abstract

This thesis first introduces some theory on coding theory and matroids, and properties that are shared between these, and then we will investigate derived matroids. In 1979 Longyear made a construction of derived matroids for binary matroids, which illuminates "dependencies among dependencies". The construction was later generalized to representable matroids by Oxley and Wang, where the derived matroid was dependent both on the matroid, and a specific representation of this matroid. The construction was generalized again by Freij-Hollandi, Jurrius and Kuznetsova to encompass all matroids. This thesis will prove that the rank of the derived matroid constructed by FJK is equal to the corank of the matroid for a large class of matroids, and the Vámos matroid is given as an example where the rank of the derived matroid is strictly smaller than the corank of the matroid. Further, some generalizations of these constructions to lattices and q-matroids are given, in addition to a generalization of the construction by Longyear to all matroids. A software library was developed alongside this thesis to calculate properties of matroids and derived matroids, and detail of this software will be given.