# A Combinatorial Methodology for Optimizing Non-Binary Graph-Based Codes: Theoretical Analysis and Applications in Data Storage

## TL;DR

This paper provides a comprehensive theoretical analysis of the weight consistency matrix (WCM) framework for optimizing non-binary LDPC codes, demonstrating significant error floor reductions and extending its application to various code types and channels.

## Contribution

It introduces a novel tree representation of GASTs, proves the optimality of the WCM framework, and extends its application to new problematic objects and code types.

## Key findings

- WCM framework is proven optimal using the unlabeled GAST tree.
- Significant error floor reductions achieved, up to 2.5 orders of magnitude.
- Extended WCM applications to various code structures and channels.

## Abstract

Non-binary (NB) low-density parity-check (LDPC) codes are graph-based codes that are increasingly being considered as a powerful error correction tool for modern dense storage devices. The increasing levels of asymmetry incorporated by the channels underlying modern dense storage systems exacerbates the error floor problem. In a recent research, the weight consistency matrix (WCM) framework was introduced as an effective NB-LDPC code optimization methodology that is suitable for modern Flash memory and magnetic recording (MR) systems. In this paper, we provide the in-depth theoretical analysis needed to understand and properly apply the WCM framework. We focus on general absorbing sets of type two (GASTs). In particular, we introduce a novel tree representation of a GAST called the unlabeled GAST tree, using which we prove that the WCM framework is optimal. Then, we enumerate the WCMs. We demonstrate the significance of the savings achieved by the WCM framework in the number of matrices processed to remove a GAST. Moreover, we provide a linear-algebraic analysis of the null spaces of WCMs associated with a GAST. We derive the minimum number of edge weight changes needed to remove a GAST via its WCMs, along with how to choose these changes. Additionally, we propose a new set of problematic objects, namely the oscillating sets of type two (OSTs), which contribute to the error floor of NB-LDPC codes with even column weights on asymmetric channels, and we show how to customize the WCM framework to remove OSTs. We also extend the domain of the WCM framework applications by demonstrating its benefits in optimizing column weight 5 codes, codes used over Flash channels with soft information, and spatially-coupled codes. The performance gains achieved via the WCM framework range between 1 and nearly 2.5 orders of magnitude in the error floor region over interesting channels.