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Let . By the Griesmer bound, for any binary, linear code. Let . Then, can be interpreted as the maximum number of occurrences of a column in the generator matrix of any code with parameters . Let be the covering radius of a [g(k, d), k, d] code. It will be shown that . Moreover, the existence of a code with is equivalent to the existence of a code. For , all codes with are described, while for a sufficient condition for their existence is formulated.