In this paper we show that APP decoding for a linear code C is optimum not for C, but for a minimum-distance-2 code which contains C as a subcode when the codewords of are transmitted with equal probability. However, APP decoding is shown to be a symptotically optimum for C for high SNR when C is a binary one-step orthogonolizable code with equiprobable codewords transmitted over the AWGN channel.
Hartmann, Carlos R.P.; Rudolph, Luther D.; Mehrotra, Kishan; and Snedeker, Guy J., "On APP Decoding" (1981). Electrical Engineering and Computer Science Technical Reports. 38.