We study the $q$-matroids associated with nondegenerate $\F_{q^m}$-linear near-MRD codes, which we call near-uniform $q$-matroids. Using the correspondence between rank-metric codes and their associated $q$-systems, we determine explicitly their rank functions and characterize their cyclic flats. We show that near-uniform $q$-matroids form a class of representable paving $q$-matroids. and prove an upper bound on the number of their cyclic flats, which is shown to be sharp for certain parameters. Finally, when $n>m$, exploiting the rank distribution of near-MRD codes, we obtain an exact count of the nontrivial cyclic flats of the associated $q$-matroids.
We show that, for any finite field $\mathbb{F}$ and positive integer $k$, a simple $\mathbb{F}$-representable matroid $M$ with sufficiently high rank contains a rank-$k$ flat which is either independent in $M$, or is a projective or affine geometry over a subfield of $\mathbb{F}$. This generalizes a result for matroids...
Jim Geelen, Matthew E. Kroeker, P. Nelson· 0 citations
Let $M$ be a binary matroid of rank $r$ with no $F_7$-minor, and let $B_M^\star$ denote the maximum of its basis-generating polynomial over nonnegative weights of total mass $r$. We prove the sharp inequality $B_M^\star\leq 2^r/(r+1)$, with equality holding if and only if the nonloop simplification of $M$ is $M(K_{r+1}...
For an integer $k$ and a set of integers $L$, a $(k,L)$-system is a family of $k$-element sets such that the size of the intersection of any two distinct members belongs to $L$. Let $m(n,k,L)$ denote the maximum size of a $(k,L)$-system on a ground set of $n$ elements. In this work, we establish a close connection betw...
We prove that the $Z$-polynomial of every sparse paving matroid has only negative real zeros, confirming the real-rootedness conjecture of Proudfoot, Xu, and Young for this class. We also prove that, for each fixed positive corank, the $Z$-polynomials of uniform matroids in consecutive ranks strictly interlace. Our app...
Let $P_q$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_q]$ over the prime field of two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the {\it hit Peterson problem} of finding a minimal set of generators for $P_q$ as a module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$...
We consider the determinant maximization problem under partition constraints: Given an $n\times n$ PSD matrix A and a partition matroid $M$ on $[n]$, find a base $S$ of $M$ that maximizes $\det(A_{S,S})$. We give an $e^{O(k)}$-approximation algorithm to find such a set $S$, where $k$ is the rank of $M$. This improves u...
Nikhil Bansal, Yu-Ze Xu· 0 citations
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