[Paper Review] On Advisability of Designing Short Length QC-LDPC Codes Using Perfect Difference Families

Alireza Tasdighi, Mohammad‐Reza Sadeghi|arXiv (Cornell University)|Jan 22, 2017

Error Correcting Code Techniques3 references3 citations

TL;DR

This paper proposes a novel construction of short-to-moderate length quasi-cyclic LDPC (QC-LDPC) codes using perfect difference families (PDFs) and a column dispersion technique (CDT) to maximize the number of short simple cycles in their Tanner graphs. Despite having relatively low minimum distances, the resulting codes achieve superior waterfall performance due to high cycle multiplicities, outperforming many existing codes in the literature while avoiding premature error floors through controlled cycle growth.

ABSTRACT

A simple and general definition of quasi cyclic low density parity check (QC LDPC) codes which are constructed based on circulant permutation matrices (CPM) is proposed. As an special case of this definition, we first represent one type of so called combinatorially designed multiple edge protograph codes. The code construction is mainly based on perfect difference families (PDF) and is called Construction 1. Secondly, using the proposed Construction 1 along with a technique named as column dispersion technique (CDT), we design several types of multiple-edge CPM QC LDPC codes (codes with Construction 2) in a wide range of rates, lengths, girths and minimum distances. Parameters of some of these codes are reported in tables. Also included in this paper are the multiplicities of short simple cycles of length up to 10 in Tanner graph of our constructed codes. Our experimental results for short to moderate length codes show that although minimum distances of codes play an important role in waterfall region, the higher the number of short simple cycles is, the better (sharper) the waterfall is. The performances of many codes such as WiMAX, PEG, array, MacKay, algebraic and combinatorial, and also, symmetrical codes have compared with our constructed codes. Based on our numerical results and regardless of how a code is constructed, those with higher number of short simple cycles and higher minimum distances, have better waterfalls.

Motivation & Objective

• To investigate the impact of short simple cycle multiplicities on the bit-error-rate (BER) performance of short-to-moderate length QC-LDPC codes.
• To address the trade-off between high cycle multiplicity (beneficial for waterfall) and error floor (detrimental), especially when minimum distance is low.
• To develop a flexible design framework that allows gradual increase in short cycle counts without triggering early error floors.
• To construct QC-LDPC codes with high performance in the waterfall region using combinatorial design based on perfect difference families (PDFs).
• To compare the performance of PDF-based codes against well-known codes (e.g., WiMAX, PEG, MacKay, symmetric) under identical parameters.

Proposed method

• A general definition of QC-LDPC codes based on circulant permutation matrices (CPMs) is introduced, enabling systematic code construction.
• Construction 1 uses perfect difference families (PDFs) to generate single-row-of-circulants QC-LDPC codes with girth 6 and minimal length.
• The column dispersion technique (CDT) is applied to Construction 1 to generate multiple-edge protograph codes with tunable rates, lengths, and girths.
• The CDT allows increasing the number of short simple cycles (especially length-6 and length-8) linearly with lifting degree N, while preserving girth and minimum distance.
• Tanner graph cycle multiplicities up to length 10 are computed and analyzed to correlate cycle counts with BER performance.
• Performance is evaluated in AWGN channels using sum-product decoding, with comparisons to benchmark codes across multiple code parameters.

Experimental results

Research questions

• RQ1 Does a high multiplicity of short simple cycles in the Tanner graph significantly improve the waterfall region of QC-LDPC codes, even when minimum distance is low?
• RQ2 Can a systematic design method be developed to increase short cycle counts without causing premature error floors?
• RQ3 How does the performance of PDF-based QC-LDPC codes compare to established codes (e.g., WiMAX, PEG, symmetric) with similar rates and lengths?
• RQ4 To what extent do minimum distance and short cycle multiplicity jointly influence BER performance in short-to-moderate length QC-LDPC codes?
• RQ5 Can the column dispersion technique (CDT) be used to flexibly control cycle counts while maintaining girth and minimum distance?

Key findings

• Codes constructed using perfect difference families (PDFs) exhibit significantly higher multiplicities of short simple cycles (especially length 6 and 8) than standard codes like WiMAX and PEG, leading to sharper waterfalls.
• The proposed code $χ^*_2$ with $N=43$ and length 570 outperforms a symmetric code at BER $=10^{-8}$ despite having a lower minimum distance, due to its much higher cycle multiplicity.
• A $(301,258)$ code $χ^*_1$ based on QPDF with $N=43$ matches the performance of a comparable CDF code from [8] down to BER $=10^{-8}$, due to nearly identical minimum distance and cycle counts.
• The column dispersion technique (CDT) successfully preserves girth and allows controlled, linear increase in short cycle multiplicities with lifting degree $N$, enabling performance tuning.
• Despite low minimum distances, PDF-based codes achieve superior waterfall performance, demonstrating that cycle multiplicity can outweigh the benefit of high minimum distance in this region.
• The results confirm that high short cycle counts improve the waterfall region, but their uncontrolled growth leads to error floors—highlighting the need for balanced design, which CDT provides.