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

Letg(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. Lets = lceil d / 2^{k-1} rceil. Then,scan 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]. Letrhobe the covering radius of a [g(k, d), k, d] code. It will be shown thatrho leq d - lceil s / 2 rceil. Moreover, the existence of a[g(k, d), k, d]code withrho = d - lceil s / 2 rceilis equivalent to the existence of a[g(k + 1, d), k + 1, d]code. Fors leq 2, all[g(k,d),k,d]codes withrho = d - lceil s / 2 rceilare described, while fors > 2a sufficient condition for their existence is formulated.