The Expected Value for the Probability of an Undetected Error

The expected value for the probability of an undetected error for a linear code over the binary symmetric channel, with p being a continuous random variable, is calculated. The random variable p is assumed to be from the interval [0,¿] or the interval [0,1], with uniform distributions and beta distributions being used. Simple conditions on the weight enumerator of the code and its dual code are given to minimize the expected value for the probability of an undetected error. A necessary condition for good error detection is derived.