Linear Programming Decoding of Spatially Coupled Codes

TL;DR

This paper proves that the LP decoding threshold for spatially coupled codes on the BSC matches that of the graph cover ensemble, revealing insights into decoding performance and dual witness properties.

Contribution

It establishes the equivalence of LP thresholds between coupled and cover ensembles and characterizes dual witness conditions with hyperflows, advancing decoding theory.

Findings

LP threshold of spatially coupled codes equals that of graph cover ensemble.

Existence of dual witness is necessary and sufficient for LP decoding success.

Derived tight bounds on hyperflow edge weights for regular LDPC codes.

Abstract

For a given family of spatially coupled codes, we prove that the LP threshold on the BSC of the graph cover ensemble is the same as the LP threshold on the BSC of the derived spatially coupled ensemble. This result is in contrast with the fact that the BP threshold of the derived spatially coupled ensemble is believed to be larger than the BP threshold of the graph cover ensemble as noted by the work of Kudekar et al. (2011, 2012). To prove this, we establish some properties related to the dual witness for LP decoding which was introduced by Feldman et al. (2007) and simplified by Daskalakis et al. (2008). More precisely, we prove that the existence of a dual witness which was previously known to be sufficient for LP decoding success is also necessary and is equivalent to the existence of certain acyclic hyperflows. We also derive a sublinear (in the block length) upper bound on the weight of any edge in such hyperflows, both for regular LPDC codes and for spatially coupled codes and we prove that the bound is asymptotically tight for regular LDPC codes. Moreover, we show how to trade crossover probability for "LP excess" on all the variable nodes, for any binary linear code.