Minimum Possible Block Length of a Linear Binary Code for Some Distances

Linear binary codes are considered. It is shown that if $d=2^{k-1}-2{k-i-1}-2^i$ or $2{k-1}-2^{k-i-1}-2^i-2$ and $k\geq2i+2$, the minimum possible block length of a code of dimension $k$ with code distance $d$ is
$$ 1=\sum^{k-1}_{j=0}\biggl\lceil\frac d{2^j}\biggr\rceil. $$