The characterization of binary constant weight codes meeting the bound of Fu and Shen

Fu and Shen gave an upper bound on binary constant weight codes. In this paper, we present a new proof for the bound of Fu and Shen and characterize binary constant weight codes meeting this bound. It is shown that binary constant weight codes meet the bound of Fu and Shen if and only if they are generated from certain symmetric designs and quasi-symmetric designs in combinatorial design theory. In particular, it turns out that the existence of binary codes with even length meeting the Grey–Rankin bound is equivalent to the existence of certain binary constant weight codes meeting the bound of Fu and Shen. Furthermore, some examples are listed to illustrate these results. Finally, we obtain a new upper bound on binary constant weight codes which improves on the bound of Fu and Shen in certain case.