This work studies the design of MET-LDPC code ensembles at low code rates with good threshold performances and exponentially few small weight codewords. There was a notable study on the condition, so called the ‘ $t$ -value condition’, for exponentially few small weight codewords. Meanwhile, it was shown that by introducing degree-one variable nodes, the threshold performances of MET-LDPC code ensembles of low rates can be significantly improved. However, the degree-one variable nodes may result in small weight codewords and thus must be carefully introduced to MET-LDPC code ensembles. Despite the importance, the existing t-value condition was developed only for MET-LDPC code ensembles without degree-one variable nodes. This work extends the t-value condition to MET-LDPC code ensembles with degree-one variable nodes, which includes the existing work as a special case. The extended t-value condition provides useful insights into the contributions of degree-one variable nodes to the distribution of small weight codewords. Thus, the results of this work allow us to design MET-LDPC code ensembles of low rates with both good threshold performances and exponentially few small weight codewords. In addition, it will be demonstrated that MET-LDPC codes at finite lengths based on the designed code ensembles have good error-rate performances both in the waterfall and low-error-rate regions.

中文翻译：

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低码率的 MET-LDPC 码集合，小权重码字呈指数级减少

这项工作研究了在低码率下具有良好阈值性能和指数级少量小权重码字的 MET-LDPC 码集合的设计。对这种情况进行了一项值得注意的研究，所谓的“ $t$ -值条件'，用于指数级的少量小权重码字。同时表明，通过引入一阶变量节点，可以显着提高低速率 MET-LDPC 码集合的阈值性能。然而，一阶变量节点可能导致小权重码字，因此必须小心地引入到 MET-LDPC 码集合中。尽管很重要，但现有的 t 值条件仅针对没有一级变量节点的 MET-LDPC 代码集合开发。这项工作将 t 值条件扩展到具有一阶变量节点的 MET-LDPC 代码集成，其中包括作为特例的现有工作。扩展的 t 值条件提供了有用的洞察力，了解度一变量节点对小权重码字分布的贡献。因此，这项工作的结果使我们能够设计具有良好阈值性能和指数级少量小权重码字的低速率 MET-LDPC 码集合。此外，将证明基于设计的代码集合的有限长度的 MET-LDPC 代码在瀑布和低错误率区域都具有良好的错误率性能。