turkmath.org

A q-polymatroid consists of a lattice of subspaces of a vector space endowed with a rank function that is both increasing and submodular. They were introduced independently by Gorla et al (2020) and Shiromoto (2019) as q-analogues of polymatroids and in reference to matrix codes. A number of invariants of codes are in fact matroid invariants, including the MacWilliams duality theorem. MacWilliams identities for classical matroids have been studied by a number of authors (e.g. Brylawski, Oxley, Britz, Shiromoto). In this talk we will consider duality of q-polymatroids and will give a version of a MacWilliams theorem for q-polymatroids, using the characteristic polynomial. As as application of this result, we will state an Assmus-Mattson-like theorem that establishes criteria for the existence of weighted subspace designs arising from a q-polymatroid. This talk is based on joint work with Michela Ceria, Relinde Jurrius, and Sorina Ionica.