Undetected error probabilities of codes for single-error correction and error detection

It is shown that the probabilities of undetected errors for maximum-length codes are upper bounded by (n+1).2^-(n-k), if the codes are used for correcting every single error and detecting other errors over a binary symmetric channel of which the transition probability is less than 1/2. Moreover, it is shown that binary linear codes with poor distance property do not satisfy the aforementioned bound.