2021/04/14 by Heide Gluesing-Luerssen, Gluesing-Luerssen, Heide, Benjamin Jany +1 · 3 citations
Engineering · #Advanced Wireless Communication Techniques #PAPR reduction in OFDM #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2104.06570
It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a q-matroid that is not representable by any linear rank-metric code and, via a relation to paving matroids, provide examples of various q-matroids that are not representable by Fqm-linear rank-metric codes. We then go on and introduce deletion and contraction for q-polymatroids and show that they are mutually dual and correspond to puncturing and shortening of rank-metric codes. Finally, we introduce a closure operator along with the notion of flats and show that the generalized rank weights of a rank-metric code are fully determined by the flats of the associated q-polymatroid.