低秩循环矩阵的构造方法及其关联的多元LDPC码

Construction of Low-rank Circulant Matrices and Their Associated Nonbinary LDPC Codes

在图像处理中,低秩矩阵的冗余信息可用于图像恢复和图像特征提取,而在迭代译码中,校验矩阵的冗余行可以加快译码收敛速度。该文研究一类易于硬件实现的低秩循环矩阵。首先将循环矩阵转换为位置集合,并基于同构理论简化了位置集合的搜索空间,从而基于比特移位方法提出了循环矩阵的构造方法。考虑非零域元素的列赋值与矩阵秩之间的关系,选取Tanner图中没有长度为4的环的循环矩阵,基于非零域元素的列赋值思想提出了不同阶数、不同码率的多元LDPC码构造方法。数值仿真结果表明,与基于PEG算法构造的二元LDPC码比较,所构造的多元LDPC码在BPSK调制方式下在误码字率10–5附近有0.9 dB的增益;在与高阶调制相结合时,有更大的性能提升。此外,所构造的多元LDPC码在迭代5次与50次下的性能几乎一致,这为低时延高可靠通信提供了一种有效的候选编码方案。

In image processing,the redundant information of low-rank matrices can be used for image recovery and image feature extraction,and redundant rows of the parity-check matrices can accelerate the convergence rate in iterative decoding.A class of low-rank circulant matrices with easy hardware implementation is studied.Circulant matrices are first converted into position sets,the search space of position sets is pruned based on isomorphism theory,and then construction of circulant matrices is proposed based on the bit shift method.Considering the relationship between the column assignment of non-zero field elements and the matrix rank,circulant matrices whose Tanner graphs have no cycles of length 4 are chosen,and according to the column assignment of non-zero field elements,construction of nonbinary LDPC codes over various finite fields and with different code rates is presented.Numerical simulation results show that,compared with binary LDPC codes constructed based on the PEG algorithm,the proposed nonbinary LDPC codes have 0.9 dB gain at Word Error Rate(WER)of 10-5 when the modulation is BPSK,and the performance gap becomes large by combining with high order modulations.Furthermore,the performance gap of the proposed codes between 5 iterations and 50 iterations is negligible,and it provides a promising coding scheme for low-latency and high-reliability communications.