Further classifications of codes meeting the Griesmer bound (Corresp.)

For any
(n, k, d)
binary linear code, the Griesmer bound says that
n \geq \sum_{i=0}^{k-1} \lceil d/2^{i} \rceil
, where
\lceil x \rceil
denotes the smallest integer
\geq x
. We consider codes meeting the Griesmer bound with equality. These codes have parameters
left( s(2^{k} - 1) - \sum_{i=1}^{p} (2^{u_{i}} - 1), k, s2^{k-1} - \sum_{i=1}^{p} 2^{u_{i} -1} \right)
, where
k > u_{1} > \cdots > u_{p} \geq 1
. We characterize all such codes when
p = 2
or
u_{i-1}-u_{i} \geq 2
for
2 \leq i \leq p
.