Linear-programming decoding of Tanner codes with local-optimality certificates

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.