On Distance Properties of Quasi-Cyclic Protograph-Based LDPC Codes

TL;DR

This paper investigates the minimum distance properties of quasi-cyclic protograph-based LDPC codes, providing bounds that account for puncturing and applying them to a standard deep-space code, enhancing understanding of their error-correcting capabilities.

Contribution

It extends existing bounds on the minimum distance of QC LDPC codes to include puncturing effects and applies these bounds to the AR4JA code used in deep-space communications.

Findings

Bounds on minimum distance are tightened for QC LDPC codes.

Puncturing effects on minimum distance are characterized.

Application of bounds to AR4JA code demonstrates practical implications.

Abstract

Recent work has shown that properly designed protograph-based LDPC codes may have minimum distance linearly increasing with block length. This notion rests on ensemble arguments over all possible expansions of the base protograph. When implementation complexity is considered, the expansion is typically chosen to be quite orderly. For example, protograph expansion by cyclically shifting connections creates a quasi-cyclic (QC) code. Other recent work has provided upper bounds on the minimum distance of QC codes. In this paper, these bounds are expanded upon to cover puncturing and tightened in several specific cases. We then evaluate our upper bounds for the most prominent protograph code thus far, one proposed for deep-space usage in the CCSDS experimental standard, the code known as AR4JA.