On the monotonic property of the probability of undetected error for a shortened code

The monotonic property of the probability of undetected error is considered when a shortened code of a linear code over GF(q) is used for error detection on a q-ary symmetric channel. Some conditions are presented under which the probability of undetected error is (or is not) monotonic with respect to the code length or the symbol error probability. It is shown that the probability of undetected error for a maximum distance separable code is monotonic with respect to the codelength. It is also shown that the probability of undetected error of a shortened code of any binary cyclic Hamming code generated by a trinomial is not monotonic with respect to the bit-error rate if the code length is short.<
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