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IEEE Convention of Electrical and Electronics Engineers in Israel

Message-passing decoding beyond the girth with local-optimality guarantees

Nissim Halabi, Guy Even

Published Nov 1, 2012• 0 citations
Publisher

Abstract

We present a new message-passing iterative decoding algorithm, called normalized weighted min-sum (NWMS). NWMS-decoding is a BP-type algorithm that applies to any irregular Tanner code with single parity-check local codes (e.g., LDPC codes and HDPC codes). The decoding guarantee of NWMS applies whenever there exists a locally optimal codeword. We prove that if a locally-optimal codeword with respect to height parameter h exists, then NWMS-decoding finds it in h iterations. This decoding guarantee holds for every finite value of h and is not limited by the girth. Because local optimality of a codeword implies that it is the unique ML-codeword, the decoding guarantee also has an ML-certificate.

DOI
10.1109/eeei.2012.6376924

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We present a new message-passing iterative decoding algorithm, called normalized weighted min-sum (NWMS). NWMS-decoding is a BP-type algorithm that applies to any irregular Tanner code with single parity-check local codes (e.g., LDPC codes and HDPC codes). The decoding guarantee of NWMS applies whenever there exists a locally optimal codeword. We prove that if a locally-optimal codeword with respect to height parameter h exists, then NWMS-decoding finds it in h iterations. This decoding guarantee holds for every finite value of h and is not limited by the girth. Because local optimality of a codeword implies that it is the unique ML-codeword, the decoding guarantee also has an ML-certificate.
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We consider decoding of binary linear Tanner codes using message-passing iterative decoding and linear-programming (LP) decoding in memoryless binary-input output-symmetric (MBIOS) channels. We present new certificates that are based on a combinatorial characterization for the local optimality of a codeword in irregular Tanner codes with respect to any MBIOS channel. This characterization is a generalization of (Arora , Proc. ACM Symp. Theory of Computing, 2009) and (Vontobel, Proc. Inf. Theory and Appl. Workshop, 2010) and is based on a conical combination of normalized weighted subtrees in the computation trees of the Tanner graph. These subtrees may have any finite height h (even equal or greater than half of the girth of the Tanner graph). In addition, the degrees of local-code nodes in these subtrees are not restricted to two (i.e., these subtrees are not restricted to skinny trees). We prove that local optimality in this new characterization implies maximum-likelihood (ML) optimality and LP optimality, and show that a certificate can be computed efficiently. We also present a new message-passing iterative decoding algorithm, called normalized weighted min-sum (NWMS). NWMS decoding is a belief-propagation (BP) type algorithm that applies to any irregular binary Tanner code with single parity-check local codes (e.g., low-density and high-density parity-check codes). We prove that if a locally optimal codeword with respect to height parameter h exists (whereby notably h is not limited by the girth of the Tanner graph), then NWMS decoding finds this codeword in h iterations. The decoding guarantee of the NWMS decoding algorithm applies whenever there exists a locally optimal codeword. Because local optimality of a codeword implies that it is the unique ML codeword, the decoding guarantee also provides an ML certificate for this codeword. Finally, we apply the new local-optimality characterization to regular Tanner codes, and prove lower bounds on the noise thresholds of LP decoding in MBIOS channels. When the noise is below these lower bounds, the probability that LP decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the Tanner graph.
This paper presents a unified analysis framework that captures recent advances in the study of local-optimality characterizations for codes on graphs. These local-optimality characterizations are based on combinatorial structures embedded in the Tanner graph of the code. Local optimality implies both unique maximum likelihood optimality and unique linear programming (LP) decoding optimality. Also, an iterative message-passing decoding algorithm is guaranteed to find the unique locally optimal codeword if one exists. We demonstrate an instance of this proof technique by considering a definition of local optimality that is based on the simplest combinatorial structures in Tanner graphs, namely, paths of length hhh. We apply the technique of local optimality to binary Tanner codes (including any low-density parity-check code, and in particular any irregular repeat-accumulate code with both even and odd repetition factors). Inverse polynomial bounds in the code length are proved on the word error probability of LP decoding for binary Tanner codes. When the local codes are restricted to single parity-check codes, these bounds also hold for decoding by a certain iterative message-passing decoding algorithm.
Given a channel observation y and a codeword x, we are interested in a one-sided error test that answers the questions: is x optimal with respect to y? is it unique? A positive answer for such a test is called a certificate for the optimality of a codeword. We present new certificates that are based on combinatorial characterization for local-optimality of a codeword in irregular Tanner codes. The certificate is based on weighted normalized trees in computation trees of the Tanner graph. These trees may have any finite height h (even greater than the girth of the Tanner graph). In addition, the degrees of local-code nodes are not restricted to two (i.e., skinny trees). We prove that local-optimality in this new characterization implies ML-optimality and LP-optimality, and show that a certificate can be computed efficiently. We apply the new local-optimality characterization to regular Tanner codes, and prove lower bounds on the noise thresholds of LP-decoding in MBIOS channels. When the noise is below these lower bounds, the probability that LP-decoding fails decays doubly exponentially in the girth of the Tanner graph.
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