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On the Growth Rate of the Weight Distribution of Irregular Doubly Generalized LDPC Codes

机译：不规则双广义LDPC码权重分布的增长率

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摘要

In this paper, the asymptotic growth rate of the weight distribution of irregular doubly generalized LDPC (D-GLDPC) codes is derived. The analysis yields a compact expression which accurately approximates the growth rate function for the case of small linear-weight codewords. This paper generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Ensembles with smallest check or variable node minimum distance greater than 2 are shown to have good growth-rate behavior, while for other ensembles a fundamental parameter is identified which discriminates between an asymptotically small and an asymptotically large expected number of small linear-weight codewords. Also, in the latter case it is shown that the growth rate depends only on the check and variable nodes with minimum distance 2. An important connection between this new result and the stability condition of D-GLDPC codes over the BEC is highlighted. Such a connection, previously observed for LDPC and GLDPC codes, is now extended to the case of D-GLDPC codes. Finally, it is shown that the analysis may be extended to include the growth rate of the stopping set size distribution of irregular D-GLDPC codes.
机译：推导了不规则双广义LDPC码（D-GLDPC）的权重分布的渐近增长率。该分析产生一个紧凑的表达式，对于线性权重小的码字，该表达式可以精确地近似增长率函数。本文概括了LDPC和广义LDPC（GLDPC）码的现有结果。具有最小校验或可变节点最小距离大于2的集合显示出具有良好的增长率行为，而对于其他集合，确定了一个基本参数，该基本参数区分渐近小的和渐进的小线性权重码字的预期数量。同样，在后一种情况下，表明增长率仅取决于距离最小为2的校验和变量节点。在BEC上，此新结果与D-GLDPC代码的稳定性条件之间的重要联系被强调。以前在LDPC和GLDPC码中观察到的这种连接现在扩展到D-GLDPC码的情况。最后，表明可以将分析扩展到包括不规则D-GLDPC码的停止集大小分布的增长率。

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