The Iby and Aladar Fleischman Faculty of Engineering

This thesis deals with two approaches to the problem of suboptimal decoding of finitelength error correcting codes under a probabilistic channel model. One approach, initiated by Gallager in 1963, is based on iterative message-passing decoding algorithms. The second approach, introduced by Feldman, Wainwright and Karger in 2003, is based on linear-programming (LP). We consider families of error correcting codes defined on graphs, called Tanner codes. The probabilistic channel model is any memoryless binaryinput output-symmetric (MBIOS) channel. 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 deal with certificates that are based on combinatorial characterizations of local-optimality. Local-optimality certificates guarantee both maximum-likelihood (ML) optimality and LP-optimality. That is, if a certificate exists for some codeword x, then LP-decoding finds x, and x is guaranteed to be the ML-codeword. We present certificates for three families of Tanner codes. (i) We start with a simple certificate based on simple cycles in the Tanner graph for a family of “even” Tanner codes. (ii) We proceed with certificates for regular low-density parity-check (LDPC) codes introduced by Arora, Daskalakis and Steurer [2009] based on a work by Koetter and Vontobel [2006]. This certificate is based on weighted skinny trees in the Tanner graph, the height of which is bounded by half of the girth of the Tanner graph. (iii) We conclude with certificate for any linear Tanner code. The certificate is based on 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., the trees are denser than skinny trees). Based on the combinatorial characterization of the certificates, we present bounds on the word error probability for LP-decoding in MBIOS channels. Upper bounds on the on the word error probability are obtained by lower bounds on the probability that a certificate exists. We present a new message-passing iterative decoding algorithm, called normalized weighted min-sum (nwms). nwms algorithm is a BP-type algorithm that applies to any Tanner code with single parity-check local codes (e.g., LDPC and high-density paritycheck codes). We prove that if a locally-optimal codeword for depth h exists, then the nwms algorithm finds it in h iterations. Hence, if successful, the nwms algorithm has an ML-certificate for any bounded number of iterations. Furthermore, since the depth h is not bounded, the guarantee for successful decoding by nwms holds even if the number of iterations h exceeds the girth of the Tanner graph. Finally, we present hierarchies on the combinatorial characterization of local-optimality. These hierarchies explain: (i) The improvement of the probability that a local-optimality certificate exists when increasing the certificate’s hight or density. (ii) The improvement of iterative decoding (e.g., nwms) when increasing number of iterations, even when the number of iterations exceeds the girth.