$k$-loose elements and $k$-paving matroids

For a matroid of rank $r$ and a non-negative integer $k$, an element is called $k$-loose if every circuit containing it has size greater than $r-k$. Zaslavsky and the author characterized all binary matroids with a $1$-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a $k$-loose element. A matroid is called $k$-paving if all its elements are $k$-loose. Rajpal showed that for a prime power $q$, the rank of a $GF(q)$-matroid that is $k$-paving is bounded. We provide a bound on the rank of $GF(q)$-matroids that are cosimple and have two $k$-loose elements. Consequently, we deduce a bound on the rank of $GF(q)$-matroids that are $k$-paving. Additionally, we provide a bound on the size of binary matroids that are $k$-paving.