On the Complexity of Exact Maximum-Likelihood Decoding for Asymptotically Good Low Density Parity Check Codes: A New Perspective

The problem of exact maximum-likelihood (ML) decoding of general linear codes is well-known to be NP-hard. In this paper, we show that exact ML decoding of a class of asymptotically good low density parity check codes — expander codes — over binary symmetric channels (BSCs) is possible with an average-case polynomial complexity. This offers a new way of looking at the complexity issue of exact ML decoding for communication systems where the randomness in channel plays a fundamental central role. More precisely, for any bit-flipping probability p in a nontrivial range, there exists a rate region of non-zero support and a family of asymptotically good codes which achieve error probability exponentially decaying in coding length n while admitting exact ML decoding in average-case polynomial time. As p approaches zero, this rate region approaches the Shannon channel capacity region. Similar results can be extended to AWGN channels, suggesting it may be feasible to eliminate the error floor phenomenon associated with belief-propagation decoding of LDPC codes in the high SNR regime. The derivations are based on a hierarchy of ML certificate decoding algorithms adaptive to the channel realization. In this process, we propose an efficient O(n^2) new ML certificate algorithm based on the max-flow algorithm. Moreover, exact ML decoding of the considered class of codes constructed from LDPC codes with regular left degree, of which the considered expander codes are a special case, remains NP-hard; thus giving an interesting contrast between the worst-case and average-case complexities

Decoder-in-the-Loop: Genetic Optimization-based LDPC Code Design

LDPC code design tools typically rely on asymptotic code behavior and are affected by an unavoidable performance degradation due to model imperfections in the short length regime. We propose an LDPC code design scheme based on an evolutionary algorithm, the Genetic Algorithm (GenAlg), implementing a "decoder-in-the-loop" concept. It inherently takes into consideration the channel, code length and the number of iterations while optimizing the error-rate of the actual decoder hardware architecture. We construct short length LDPC codes (i.e., the parity-check matrix) with error-rate performance comparable to, or even outperforming that of well-designed standardized short length LDPC codes over both AWGN and Rayleigh fading channels. Our proposed algorithm can be used to design LDPC codes with special graph structures (e.g., accumulator-based codes) to facilitate the encoding step, or to satisfy any other practical requirement. Moreover, GenAlg can be used to design LDPC codes with the aim of reducing decoding latency and complexity, leading to coding gains of up to $0.325$ dB and $0.8$ dB at BLER of $10^{-5}$ for both AWGN and Rayleigh fading channels, respectively, when compared to state-of-the-art short LDPC codes. Also, we analyze what can be learned from the resulting codes and, as such, the GenAlg particularly highlights design paradigms of short length LDPC codes (e.g., codes with degree-1 variable nodes obtain very good results).Comment: in IEEE Access, 201

An Iterative Joint Linear-Programming Decoding of LDPC Codes and Finite-State Channels

In this paper, we introduce an efficient iterative solver for the joint linear-programming (LP) decoding of low-density parity-check (LDPC) codes and finite-state channels (FSCs). In particular, we extend the approach of iterative approximate LP decoding, proposed by Vontobel and Koetter and explored by Burshtein, to this problem. By taking advantage of the dual-domain structure of the joint decoding LP, we obtain a convergent iterative algorithm for joint LP decoding whose structure is similar to BCJR-based turbo equalization (TE). The result is a joint iterative decoder whose complexity is similar to TE but whose performance is similar to joint LP decoding. The main advantage of this decoder is that it appears to provide the predictability of joint LP decoding and superior performance with the computational complexity of TE.Comment: To appear in Proc. IEEE ICC 2011, Kyoto, Japan, June 5-9, 201