**Published at : ** 30 Oct 2019

**IJtech :** IJtech
Vol 10, No 5 (2019)

**DOI :** https://doi.org/10.14716/ijtech.v10i5.1144

Amine, T.M., Ali, D. 2019. A New Method for Building Low-Density-Parity-Check Codes.

30

Tehami Mohammed Amine | Telecommunications and Digital Signal Processing Laboratory, Djillali liabes University of Sidi Bel Abbes, Algeria |

Djebbari Ali | Telecommunications and Digital Signal Processing Laboratory, Djillali liabes University of Sidi Bel Abbes, Algeria |

Abstract

This paper proposes a new method for building
low-density-parity-check codes, exempt of cycle of length 4, based on a
circulant permutation matrix, which needs very little memory for storage it in
the encoder and a dual diagonal structure is applied to guarantee that parity
check bits can be recursively computed with linear calculation complexity. The
Bit Error Rate performance of the new low-density-parity-check codes was
compared to the uncoded bi-phase-shift-keying over
additive-white-gaussian-noise channel. This simulation shows that the proposed
codes are very efficient over additive-white-gaussian-noise. The proposed codes
ensure a very low encoding complexity and reduce the memory storage required
for the parity-check matrix, which can be more easily built than others codes
used in channel coding.

Circulant permutation matrix; Dual-diagonal matrix; Girth; Low-density-parity-check codes; Parity-check matrix

Introduction

Because of their prodigious performance, low-density-parity-check (LDPC)
codes are now considered optimal (Gallager, 1962). These codes are a part of
linear block codes which have acquired considerable importance in error
correcting performances (Yahya et al., 2009). LDPC codes can be presented by specific parity check matrix H
that includes a high density of 0’-s and a low density of 1’-s (Mackay, 1999).
The Tanner graph (Tanner, 1981) is a bipartite graph comprising two groups of
nodes: the variable nodes and check nodes. Variable nodes depicting columns and
rows are represented by check nodes and connections between these two sets are
known as edges (Tanner, 1981; Juwono et al., 2013). In the Tanner graph, a
cycle is defined as a path which starts and ends at the same node, if the graph
contains a cycle; its minimum length is known as ‘girth’ (Tanner, 1981). Cycles
especially those of length 4 decrease the bit-error-rate (BER) performance of
LDPC codes, because of their impact on the independence of extrinsic
information exchanged in the decoding process (Johnson & Weller, 2001).
Gallager codes are classified: as regular if the weight of columns and rows
(i.e. density of 1’s) is constant and as irregular if column weight and row
weight are variable (Yahya et al., 2009).

The
construction of these codes is of two types; the first is random construction
that is flexible in design and construction (Mackay, 1999). The parity-check
matrix is a superposition and/or concatenation of sub-matrices and this
construction has significant drawbacks in term of the stocking and accessing a
large parity-check matrix. As random building does not guarantee small cycle
lengths, a second form of construction was developed; this is known as
deterministic construction (Moura el al., 2004; Shin et al., 2014).

In this paper, we depict a particular
category of LDPC codes, excluding cycles of length 4, which can be linearly
coded by matrix H. The parity check matrix is divided into two sections: the
first, which matches to the parity bits, is a dual-diagonal structure (Guolei
& Dong, 2010) and the second, which matches the information bits, is a
quasi-cyclic structure. For that reason, this new LDPC code is classified as
irregular code.

This paper is organized as follows. Section 2 discusses the construction of the new LDPC code based on a quasi-cyclic and a dual diagonal matrix. In section 3, we propose a deterministic rule for constructing parity-check matrix with various rates. Section 4 describes the LDPC reduced-complexity encoding method, and section 5 discusses decoding complexity. Section 6 specifies the advantages of the proposed method, followed by conclusions in Section 7.

Conclusion

To address quality of reception and implementation
constraints, LDPC code must be constructed with a low error floor, linear
encoding and less complex decoding. This
paper proposes a new method for constructing parity-check matrix that include girths of length 4, for different rates. Memory requirements are significantly reduced by the use of the quasi-cyclic
matrices and
dual- diagonal, which reduce encoding complexity.

Supplementary Material

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R2-EECE-1144-20191015090437.docx | --- |

R2-EECE-1144-20191015090451.docx | --- |

R2-EECE-1144-20191015090501.docx | --- |

R2-EECE-1144-20191015090512.docx | --- |

R2-EECE-1144-20191015090531.docx | --- |

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