Binary set functions and parity check matrices

We consider the possibility of extending to a family of sets a binary set function defined on a subfamily so that the extension is, in fact, uniquely determined. We place in this context the problem of finding the least integer n(r) such that every linear code of length n with n B n(r), dimension n r and minimum Hamming distance at least 4 has a parity check matrix composed entirely of odd weight columns and answer this problem by showing that n(r) = 5 . 2’-4 + 1, r 2 4. This result is applied to yield new constructions and bounds for unequal error protection codes with minimum distances 3 and 4.