The Lattice of Cyclic Flats of a Matroid

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice Z of sets and a function r on Z to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to Z. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids.Comment: 15 pages, 1 figure. The new version addresses earlier work by Julie Sims that the authors learned of after submitting the first versio

Gammoids and transversal matroids

AbstractThe two main results of this paper identify the “strict gammoids” of Mason [7] with duals of transvesal matroids, and gammoids in general with contractions of transversal matroids. Both theorems derive from a fundamental construction which we also use, inter alia, to establish a duality between the graph theorems of Menger and König

On the Combinatorics of Locally Repairable Codes via Matroid Theory

This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters $(n,k,d,r,\delta)$ of LRCs are generalized to matroids, and the matroid analogue of the generalized Singleton bound in [P. Gopalan et al., "On the locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes, that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters $(n,k,d,r,\delta)$ and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given in [W. Song et al., "Optimal locally repairable codes," IEEE J. Sel. Areas Comm.].Comment: 48 pages. Submitted for publication. In this version: The text has been edited to improve the readability. Parameter d for matroids is now defined by the use of the rank function instead of the dual matroid. Typos are corrected. Section III is divided into two parts, and some numberings of theorems etc. have been change

Presentations of transversal valuated matroids

Once this article is published by the JLMS, the Elements version will need to bear a statement of the following form: "This is the accepted version of the following article: FULL CITE, which has been published in final form at [Link to final article]"Once this article is published by the JLMS, the Elements version will need to bear a statement of the following form: "This is the accepted version of the following article: FULL CITE, which has been published in final form at [Link to final article]