On optimal binary codes with unbalanced coordinates

A coordinate of a binary code of size M is said to be balanced if the number of zero and ones in the coordinate is either or (that is, exactly for even M). Since good codes (of various types) tend to be balanced in all coordinates, various conjectures have been made regarding the existence of such codes. It is here shown that there are parameters for which there are no optimal binary error-correcting codes with a balanced coordinate. This is proved by the code attaining , which is shown to be unique here; denotes the maximum size of a binary code of length n and minimum distance d. It is further shown that .