Instanton search algorithm for the ADMM penalized decoder

Linear programming (LP) decoding using the alternating direction method of multipliers (ADMM) has been shown to be an efficient algorithm. A non-convex variation based on the ADMM LP decoder called the ADMM penalized decoder was introduced by Liu et al. (IEEE ITW, Sep. 2012) to close the signal-to-noise ratio (SNR) gap between LP decoding and classic belief propagation (BP) decoding. This algorithm was shown to achieve or outperform BP decoding at all SNRs, including high SNRs where BP decoding suffers from the error floor effect. In this paper, we study the behaviors of the ADMM penalized decoder at high SNRs where simulation is infeasible. We use a generic tool called instanton analysis and propose an instanton search algorithm for the ADMM penalized decoder. We then apply the algorithm to the [155, 64] Tanner code and a [1057, 813] LDPC code. We show that the instanton information we obtained provides good predictions for word-error-rate curve at high SNRs. In addition, our results suggest that the ADMM penalized decoder can suffer from trapping sets.