Undetected error probability of hamming code for any number of symbols

In past papers, it has been shown that probability of undetected error P_u(ε) for binary (n = 2^m - 1, k = n - m, 3) Hamming code (q = 2) used for error detection on binary symmetric channel satisfies the 2^-p bound, where p is the parity check bits equal to n - k, hence binary Hamming codes are proper. In this correspondence this result is generalized and it has been shown that not only binary but Hamming Codes (for any value of q) satisfy this bound, so generalized Hamming codes are proper.