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We study the stability of low-density parity-check (LDPC) codes underblockwise or bitwise maximum $\textit{a posteriori}$ (MAP) decoding, wheretransmission takes place over a binary-input memoryless output-symmetricchannel. Our study stems from the consideration of constructing universalcapacity-achieving codes under low-complexity decoding algorithms, whereuniversality refers to the fact that we are considering a family of channelswith equal capacity. Consider, e.g., the right-regular sequence by Shokrollahiand the heavy-tail Poisson sequence by Luby $\textit{et al}$. Both sequencesare provably capacity-achieving under belief propagation (BP) decoding whentransmission takes place over the binary erasure channel (BEC).  In this paper we show that many existing capacity-achieving sequences of LDPCcodes are not universal under BP decoding. We reveal that the key to showingthis non-universality result is determined by the stability of the underlyingcodes. More concretely, for an ordered and complete channel family and asequence of LDPC code ensembles, we determine a stability threshold associatedwith them, which further gives rise to a sufficient condition such that thesequence is not universal under BP decoding. Furthermore, we show that the samestability threshold applies to blockwise or bitwise MAP decoding as well. Wepresent how stability can determine an upper bound on the correspondingblockwise or bitwise MAP threshold, revealing the operational significance ofthe stability threshold.

The Stability of Low-Density Parity-Check Codes and Some of Its Consequences