# Pseudocodeword-Free Criterion for Codes with Cycle-Free Tanner Graph

## TL;DR

This paper establishes a criterion for LDPC codes with cycle-free Tanner graphs, showing that cycles in redundant rows do not affect error performance, and characterizes all pseudocodeword-free parity-check matrices for these codes.

## Contribution

It introduces a pseudocodeword-free criterion for codes with cycle-free Tanner graphs and characterizes all such parity-check matrices, clarifying the role of cycles in error performance.

## Key findings

- Cycles in redundant rows do not impact error performance.
- Characterization of all pseudocodeword-free parity-check matrices.
- Cycle-free Tanner graphs guarantee convergence to maximum-likelihood decoding.

## Abstract

Iterative decoding and linear programming decoding are guaranteed to converge to the maximum-likelihood codeword when the underlying Tanner graph is cycle-free. Therefore, cycles are usually seen as the culprit of low-density parity-check (LDPC) codes. In this paper, we argue in the context of graph cover pseudocodeword that, for a code that permits a cycle-free Tanner graph, cycles have no effect on error performance as long as they are a part of redundant rows. Specifically, we characterize all parity-check matrices that are pseudocodeword-free for such class of codes.