Good error-detection codes satisfying the expurgated bound

A q-ary (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 exactly by, for example, all the binary perfect codes, binary first-order Reed-Muller codes, binary (extended) Hamming code, binary 2-error correcting primitive BCH code, and all the maximum distance separable (MDS) codes, and asymptotically by some class of the binary t-error-correcting primitive BCH codes