# The Velocity of the Decoding Wave for Spatially Coupled Codes on BMS Channels

## TL;DR

This paper analyzes the wave-like propagation of decoding errors in spatially coupled LDPC codes over BMS channels, deriving a formula for the wave velocity and supporting it with numerical evidence.

## Contribution

It provides a theoretical formula for the velocity of the decoding wave in spatially coupled codes and introduces practical approximations validated by numerical data.

## Key findings

- Derived a formula for the decoding wave velocity v
- Proposed simpler approximations for v
- Supported findings with numerical data

## Abstract

We consider the dynamics of belief propagation decoding of spatially coupled Low-Density Parity-Check codes. It has been conjectured that after a short transient phase, the profile of "error probabilities" along the spatial direction of a spatially coupled code develops a uniquely-shaped wave-like solution that propagates with constant velocity v. Under this assumption, and for transmission over general Binary Memoryless Symmetric channels, we derive a formula for v. We also propose approximations that are simpler to compute and support our findings using numerical data.