^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

We present an approach for constructing LDPC codes without cycles of length 4 and 6. Firstly, we design 3 submatrices with different shifting functions given by the proposed schemes, then combine them into the matrix specified by the proposed approach, and, finally, expand the matrix into a desired parity-check matrix using identity matrices and cyclic shift matrices of the identity matrices. The simulation result in AWGN channel verifies that the BER of the proposed code is close to those of Mackay's random codes and Tanner's QC codes, and the good BER performance of the proposed can remain at high code rates.

LDPC codes can
be described by a bipartite graph called Tanner graph [

This letter is
structured as follows. In Section

Section

An

A cycle also
can be shown in a tree. Figure

A 4-cycle in tree.

A 6-cycle in tree.

In Figure

A 4-cycle in a parity-check matrix.

Figure

A 6-cycle in a parity-check matrix.

From Figure

Six kinds of 6-cycles with different figures.

Figure

If there is a
parity-check matrix

The design algorithm of

Design a matrix

Let

Let

The design algorithm of

Design a
matrix

Let

Let

Let

The design algorithm of

Designing a matrix

Let

Let

Let

We can get

The expansion algorithm is as follows.

Select an identity matrix

Let

Exchanging the 1s in

From the structure of the parity-check matrix

In Section

If a matrix

We can know from Figure

In the
submatrix

If a matrix

As shown in Figure

In the
submatrix

We know that

If there is no 4-cycle and 6-cycle in
the matrix

If there is a 4-cycle

To demonstrate the error-correcting
performance, we constructed two rate-1/2 LDPC codes by the proposed method. For the purpose of comparison, we also construct
two classes of Tanner’s
QC codes and Mackay’s random codes [

Parameters of example codes with rate-1/2.

6 | 3 | 1080 | 1/2 | 8 | |

6 | 3 | 1008 | 1/2 | 6 | |

6 | 3 | 1002 | 1/2 | 8 |

Table

Parameters of our codes with high-rate.

9 | 3 | 5103 | 0.67 | 8 | |

10 | 3 | 5000 | 0.7 | 8 | |

12 | 3 | 5184 | 0.75 | 8 |

We simulate
the proposed code’s error-correcting performance with the assumption that each
code is modulated by BPSK and transmitted over additive white Gaussian noise
(AWGN) channel. All the codes are decoded with the sum-product algorithm [

BER performance comparison of our code, Mackay’s random code and Tanner’s QC code.

BER performance of our codes at high code rate.

In this paper,
we proposed a QC LDPC code without girth-4 and girth-6, three lemmas are
provided to prove the short girths’ properties of the proposed codes.
Simulation verified the good error-correcting performance of the proposed code,
whose BER performance is near to those of Tanner’s QC codes and MacKay’s random
codes [

This paper is supported by the National Natural Science Foundation of China under Grant no. 60572093, Specialized Research Fund for the Doctoral Program of Higher Education (20050004016) of China, and IITA Professorship Program of Gwangju Institute of Science and Technology, South Korea. Part of contents of this paper was presented on IET International Conference on Wireless Mobile and Multimedia Networks Proceedings (ICWMMN 2006).