Good error detection codes satisfy the expurgated bound

A q-nary (n,k) linear code is said to be proper if, as an error-detection code, the probability of undetectable error, P_ud , satisfies P_ud⩽q^-(n-k) for completely symmetric channels. We show that a proper code, as an error-correction code, satisfies the expurgated bound on the decoding error probability for a class of channels with the associated Bhattacharyya distance being completely symmetric. Known results on the undetectable error probability then immediately imply that the expurgated exponent is satisfied by many codes which are regarded as good codes