Polytope of Correct (Linear Programming) Decoding and Low-Weight Pseudo-Codewords

TL;DR

This paper studies LP decoding of binary codes, showing the error surface as a polytope, and introduces new heuristics for finding low-weight pseudo-codewords, with tests on a Tanner code over AWGN.

Contribution

It formulates the problem of finding lowest-weight pseudo-codewords as a non-convex optimization over a polytope, leading to new heuristics with provable convergence.

Findings

Error surface is a polytope in LP decoding.

New heuristics improve the search for low-weight pseudo-codewords.

Algorithm tested successfully on Tanner code over AWGN.

Abstract

We analyze Linear Programming (LP) decoding of graphical binary codes operating over soft-output, symmetric and log-concave channels. We show that the error-surface, separating domain of the correct decoding from domain of the erroneous decoding, is a polytope. We formulate the problem of finding the lowest-weight pseudo-codeword as a non-convex optimization (maximization of a convex function) over a polytope, with the cost function defined by the channel and the polytope defined by the structure of the code. This formulation suggests new provably convergent heuristics for finding the lowest weight pseudo-codewords improving in quality upon previously discussed. The algorithm performance is tested on the example of the Tanner [155, 64, 20] code over the Additive White Gaussian Noise (AWGN) channel.