This paper analyzes and explicitly constructs quasi-cyclic (QC) codes for correcting multiple bursts via matrix transformations. Our analysis demonstrates that the multiple-burst-correction capability of QC codes is determined by sub-matrices in the diagonal of their transformed parity-check matrices. By well designing these sub-matrices, the proposed QC codes are able to achieve optimal or asymptotically optimal multiple-burst-correction capability. Moreover, it proves that these codes can be QC low-density parity-check (QC-LDPC) codes, if the diagonal sub-matrices of their transformed parity-check matrices are Hadamard powers of base matrices. Analysis and simulation results show that our QC-LDPC codes perform well over not only random symbol error/erasure channels, but also burst channels.

中文翻译：

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变换域中多重突发校正码的构建及其与LDPC码的关系

本文分析并显式构建准循环 (QC) 码，用于通过矩阵变换校正多个突发。我们的分析表明，QC 码的多重突发校正能力是由其变换奇偶校验矩阵对角线上的子矩阵决定的。通过很好地设计这些子矩阵，所提出的 QC 代码能够实现最佳或渐近最佳的多次突发校正能力。此外，它证明了这些代码可以是QC低密度奇偶校验（QC-LDPC）码，如果它们变换的奇偶校验矩阵的对角子矩阵是基矩阵的Hadamard幂。分析和仿真结果表明，我们的 QC-LDPC 码不仅在随机符号错误/擦除信道上表现良好，而且在突发信道上也表现良好。