Local Optimality Certificates for LP Decoding of Tanner Codes

TL;DR

This paper introduces a novel combinatorial characterization for local optimality in LP decoding of Tanner codes, based on linear combinations of subtrees, which can improve decoding bounds and be efficiently computed.

Contribution

It proposes a new local optimality criterion using subtrees with arbitrary degrees and heights, enhancing decoding analysis and certification methods.

Findings

New characterization implies ML and LP optimality.

Efficient algorithms for local optimality certification.

Potential for improved decoding bounds.

Abstract

We present a new combinatorial characterization for local optimality of a codeword in an irregular Tanner code. The main novelty in this characterization is that it is based on a linear combination of subtrees in the computation trees. These subtrees may have any degree in the local code nodes and may have any height (even greater than the girth). We expect this new characterization to lead to improvements in bounds for successful decoding. We prove that local optimality in this new characterization implies ML-optimality and LP-optimality, as one would expect. Finally, we show that is possible to compute efficiently a certificate for the local optimality of a codeword given an LLR vector.