Analysis of Spatially-Coupled Counter Braids

TL;DR

This paper analyzes spatially-coupled counter braids using an equivalent bipartite graph model, revealing thresholds related to decoding performance and conjecturing their equality with the Maxwell decoding threshold.

Contribution

It introduces an equivalent graph representation for CBs and SC-CBs, establishing threshold relationships and conjecturing their connection to the MAP decoding limit.

Findings

Potential and area thresholds are equal in the equivalent graph.

The area under the BP extrinsic information transfer curve is zero at the area threshold.

Numerical results suggest a gap between BP decoding threshold and the conjectured MAP threshold.

Abstract

A counter braid (CB) is a novel counter architecture introduced by Lu et al. in 2007 for per-flow measurements on high-speed links. CBs achieve an asymptotic compression rate (under optimal decoding) that matches the entropy lower bound of the flow size distribution. Spatially-coupled CBs (SC-CBs) have recently been proposed. In this work, we further analyze single-layer CBs and SC-CBs using an equivalent bipartite graph representation of CBs. On this equivalent representation, we show that the potential and area thresholds are equal. We also show that the area under the extended belief propagation (BP) extrinsic information transfer curve (defined for the equivalent graph), computed for the expected residual CB graph when a peeling decoder equivalent to the BP decoder stops, is equal to zero precisely at the area threshold. This, combined with simulations and an asymptotic analysis of the Maxwell decoder, leads to the conjecture that the area threshold is in fact equal to the Maxwell decoding threshold and hence a lower bound on the maximum a posteriori (MAP) decoding threshold. Finally, we present some numerical results and give some insight into the apparent gap of the BP decoding threshold of SC-CBs to the conjectured lower bound on the MAP decoding threshold.