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International Conference on Engineering and Telecommunication (EnT)

On the Nested Family of LDPC Codes Based on Golomb Rulers

Fedor I. Ivanov, Pavel S. Rybin

Published Nov 1, 2017• 2 citations
Publisher

Abstract

In this paper we present a new ensemble of Low- Density Parity-Check codes (LDPC Codes) based on modular Golomb rulers and permutation matrices. We suggest a designing rule for these codes that allows obtaining a nested family of LDPC codes with wide range of rates from one matrix of lowest rate code. Moreover, the regular way that we apply to construct our codes allows describing parity-check matrices with minimal number of parameters. The simulation results in the case of transmission via AWGN channel with QAM-4 modulation are presented.

DOI
10.1109/icent.2017.21

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In this paper we present a new ensemble of Low- Density Parity-Check codes (LDPC Codes) based on modular Golomb rulers and permutation matrices. We suggest a designing rule for these codes that allows obtaining a nested family of LDPC codes with wide range of rates from one matrix of lowest rate code. Moreover, the regular way that we apply to construct our codes allows describing parity-check matrices with minimal number of parameters. The simulation results in the case of transmission via AWGN channel with QAM-4 modulation are presented.
Low-density lattice codes (LDLC) are a new type of lattice codes that can be decoded effectively and achieve performance levels near the capacity of the additive white Gaussian noise (AWGN) channel. Earlier designs of LDLC, such as those utilizing Latin squares, relied on high-complexity computer searches to eliminate short cycles. The proposed code construction offers a systematic representation and inherently avoids these issues. This study presents a design for LDLC that is based on Golomb rulers. This approach is deterministic and results in a parity-check matrix that is free of short cycles. In all scenarios examined, the LDLC built on Golomb rulers demonstrated superior frame error rate performance compared to those based on Latin squares. For instance, there is a 1.5 dB improvement observed for a dimension ofn=73\mathrm{n}=73n=73.
We introduce and study a family of rate-compatible Low-Density Parity-Check (LDPC) codes. The design of these codes starts from simplex codes, defined by parity-check matrices having a simple form stemming from the coefficients of a primitive polynomial. For this reason, we call the new codes Primitive Rate-Compatible LDPC (PRC-LDPC) codes. By applying puncturing to these codes, we obtain a bit-level granularity of the code rate. We show that, in order to achieve good LDPC codes, the underlying polynomials, besides being primitive, must meet some more stringent conditions with respect to those of classical punctured simplex codes. We leverage non-modular Golomb rulers to take these new requirements into account. We characterize the minimum distance properties of PRC-LDPC codes, and study and discuss their encoding and decoding complexity. Finally, we assess the error rate performance of high rate PRC-LDPC codes under iterative decoding.
Based jointly on idempotents and modular Golomb rulers, we construct a class of nonbinary cyclic low-density parity-check (LDPC) codes. The defining parity-check matrix is a sparse circulant, on which we put two constraints: 1) the characteristic polynomial is an idempotent, 2) the nonzero elements of the first row are located on a modular Golomb ruler. We show that the second constraint forms a necessary and sufficient condition for the Tanner graph to have no cycles of length 4. The minimum distance of the code is proved equal to the column weight of the parity-check matrix plus one. A search algorithm is presented, with which we obtain some high rate codes with large minimum distances. The issue of code equivalence is also discussed. Simulation results show that the obtained codes perform well under iterative decoding.
In this study, a new method for constructing low-density parity-check (LDPC) codes is presented. This construction is based on permutation matrices which come from a finite abstract group and hence the codes constructed in this manner are called group permutation low-density parity-check (GP-LDPC) codes. A necessary and sufficient condition under which a GP-LDPC code has a cycle is given and some properties of these codes are investigated. A class of flexible-rate GP-LDPC codes without cycles of length four is also introduced. Simulation results show that GP-LDPC codes perform very well with the iterative decoding and can outperform their random-like counterparts.
A design algorithm of regular check matrix for Low-Density Parity-Check(LDPC)codes is proposed in this paper.The proposed approach is to design 3 sub-matrices,and each of the sub-matrices is combined by the cyclic shift matrices of the identity matrix,then to combine these sub-matrices into a desired parity check matrix according to our approach.Using the check algorithms of the short cycles to find the combinations of shifting operators by which the parity check matrix desires has no girth- four and no girth-six.The simulation results in AWGN channel show that the BER performance of these codes obtained by the generator matrix derived from the proposed check matrices to encode the random information bits is as good as that of random LDPC codes.