q-polymatroids and their relation to rank-metric codes

It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously matroid theory, one may ask whether a given q-polymatroid representable by code. We provide an answer presenting example of q-matroid not any code and, via relation paving matroids, examples various q-matroids are $${{\mathbb {F}}}_{q^m}$$ -linear codes. then go on and introduce deletion contraction for q-polymatroids show they mutually dual correspond puncturing shortening Finally, we closure operator along with the notion flats generalized rank weights fully determined associated q-polymatroid.