On the covering radius of binary, linear codes meeting the Griesmer bound

Let g(k, d) = \sum_{i=0}^{k-1} \lceil d / 2^{i} \rceil . By the Griesmer bound, n \geq g(k, d) for any binary, linear [n, k, d] code. Let s = \lceil d / 2^{k-1} \rceil . Then, s can be interpreted as the maximum number of occurrences of a column in the generator matrix of any code with parameters [g(k, d), k, d] . Let \rho be the covering radius of a [g(k, d), k, d] code. It will be shown that \rho \leq d - \lceil s / 2 \rceil . Moreover, the existence of a [g(k, d), k, d] code with \rho = d - \lceil s / 2 \rceil is equivalent to the existence of a [g(k + 1, d), k + 1, d] code. For s \leq 2 , all [g(k,d),k,d] codes with \rho = d - \lceil s / 2 \rceil are described, while for s > 2 a sufficient condition for their existence is formulated.