Half Rate Quasi Cyclic Low Density Parity Check Codes Based on Combinatorial Designs

Narges Rezvani Majid,Sina Vafi

doi:10.4236/jcc.2016.412002

Abstract

This paper presents new half rate Quasi Cyclic Low Density Parity Check (QC- LDPC) codes formed on the basis of combinatorial designs. In these codes, circulant matrices of the parity check matrix are formed on the basis of subsets in which the difference between any two elements of a subset is unique with all differences obtained from the same or different subsets. This structure of circulant matrices guarantees non-existence of cycle-4 in the Tanner graph of QC-LDPC codes. First, an irregular code with girth 6 constituted by two rows of circulant matrices is proposed. Then, more criteria will be considered on the structure of subsets with the mentioned feature aiming to represent a new scheme of regular QC-LPDC codes with girth at least 8. From simulations, it is confirmed that codes have similar to or better performance than other well-known half rate codes, while require lower complexity in their design.

Highlights

Quasi-Cyclic Low Density Parity Check (QC-LDPC) codes are represented as reputable structured-type LDPC codes, which are considered in the current and generations of broadband transmission and storage systems [1] [2]
This paper presents new half rate Quasi Cyclic Low Density Parity Check (QCLDPC) codes formed on the basis of combinatorial designs
By the same argument presented in Lemma 1, it is possible to have other conditions for the existence of a cycle-4, which are dependent on elements of subsets applied in construction of parity check matrix of QC-LDPC code

Summary

Introduction

Quasi-Cyclic Low Density Parity Check (QC-LDPC) codes are represented as reputable structured-type LDPC codes, which are considered in the current and generations of broadband transmission and storage systems [1] [2]. CPM-based parity check matrix of QC-LDPC codes is possibly formed by combination of finite fields and combinatorial designs In this case, circulant matrices are obtained by combination of two arbitrary subsets of elements from a defined field. Despite the method presented in [13], codes have half rate and subsets are defined by unequal lengths Based on this feature of circulant matrices, an irregular QC-LDPC code with girth 6 is proposed. Instead of utilizing a masking technique, circulant matrices are interactively designed with each other to prohibit existence of cycles with lengths 4 and 6 in the Tanner graph of the parity check matrix.

Subsets with Different Lengths and Unique Differences between Elements

Irregular Half Rate QC-LDPC Codes with Girth 6

Regular Half Rate QC-LDPC Codes with Girth 8

Simulation Results

Conclusions and Future Work