An Improvement of Greismer Bound for Some Classes of Distances

Linear binary codes are considered. We prove that for $d=2^{k-2}-2^{а}-2^{Ь}$, $0\leq b<a\leq k-3$, $2\leq a$, $9\leq k$, the block length of a $k$-dimensional code with code distance $d$ is not less than
$$ 2+\sum_{j=0}^{k-1}\lceil\frac{d}{2^j}\rceil. $$