$K_1$-groups via binary complexes of fixed length

We modify Grayson's model of $K_1$ of an exact category to give a presentation whose generators are binary acyclic complexes of length at most $k$ for any given $k \ge 2$. As a corollary, we obtain another, very short proof of the identification of Nenashev's and Grayson's presentations.
Comment: 10 pages, minor changes following a referee report, to appear in HHA