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-po...

We demonstrate that two key theorems of Amaldi, Liberti, Maffiolo and Maculan (2009), which they presented with rather complicated proofs, can be more easily and cleanly established using a simple and classical property of binary matroids. Besides a simpler proof, we see that both of these key results are manifestations of the same essential property. Our goal is to demonstrate that two graph-t...

Subset selection problems ask for a small, diverse yet representative subset of the given data. When pairwise similarities are captured by a kernel, the determinants of submatrices provide a measure of diversity or independence of items within a subset. Matroid theory gives another notion of independence, thus giving rise to optimization and sampling questions about Determinantal Point Processe...

The aim of this paper is to develop, by analogy with the theory of oriented (ordinary) matroids [2], an oriented version of the theory of Lagrangian symplectic matroids [7]. Recall that the concept of an oriented matroid axiomatises, in combinatorial form, the properties of the ensemble of non-zero k× k minors of a real k× n matrix A, say. The k-subsets of indices of columns of A such that the ...

Matroids are important combinatorial structures and connect close-lywith graphs. Matroids and graphs were all generalized to fuzzysetting respectively. This paper tries to study  connections betweenfuzzy matroids and fuzzy graphs. For a given fuzzy graph, we firstinduce a sequence of matroids  from a sequence of crisp graph, i.e.,cuts of the fuzzy graph. A fuzzy matroid, named graph fuzzy matro...

Now, we come to offer you the right catalogues of book to open. a source book in matroid theory is one of the literary work in this world in suitable to be reading material. That's not only this book gives reference, but also it will show you the amazing benefits of reading a book. Developing your countless minds is needed; moreover you are kind of people with great curiosity. So, the book is v...

Let two linear matroids have the same rank in matroid intersection. A maximum linear matroid intersection (maximum linear matroid parity set) is called a basic matroid intersection (basic matroid parity set), if its size is the rank of the matroid. We present that enumerating all basic matroid intersections (basic matroid parity sets) is in NC, provided that there are polynomial bounded basic m...

We establish a formalism for working with incidence algebras of posets symmetries, and we develop equivariant Kazhdan–Lusztig–Stanley theory within this formalism. This gives new way thinking about the Kazhdan–Lusztig polynomial Z-polynomial matroid.

Index 74 Preface The three subjects of the title (codes, matroids, and permutation groups) have many interconnections. In particular, in each case, there is a polynomial which captures a lot of information about the structure: we have the weight enumerator of a code, the Tutte polynomial (or rank polynomial) of a matroid, and the cycle index of a permutation group. With any code is associated a...