The undetected error probabilities of combinatorial codes and their dual codes

IT is known that the average undetected error probability (UEP) of a binary [n, k] code Con a binary symmetric channel with crossover probability p is given byP(p) = sum from i=1 Ap~i(1-p), (1)where (A, A,…, A) is the weight distribution of C. If for all 0≤p≤0.5, P(p)≤P(0. 5) = 2-2, then C is called good for error detection. Moreover, if P(p) ismonotonously increasing in the interval [0, 0.5], then C is called proper. Clearly, propercodes are good.