Some Intersections of two Binary LRCs with Disjoint Repair Groups

In this paper, we investigate some properties of inter-sections of two binary
$(n, k, r)$
LRCs with disjoint repair groups when
$m\triangleq\frac{n}{r+1}$
is an integer. We first prove a necessary and sufficient condition for such an intersection to have availability 2. We prove as a corollary that such an intersection with availability 2 must be in fact a 3-seq LRC. Furthermore, we prove a bound on the dimension of such intersections and identify an interesting relation between this and the general rate bound on 3-seq LRCs.