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Bounds on the Number of Iterations for Turbo-Like Ensembles Over the Binary Erasure Channel

机译：二进制擦除通道上类似Turbo的乐团的迭代次数的界

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摘要

This paper provides simple lower bounds on the number of iterations which is required for successful message-passing decoding of some important families of graph-based code ensembles (including low-density parity-check (LDPC) codes and variations of repeat–accumulate codes). The transmission of the code ensembles is assumed to take place over a binary erasure channel, and the bounds refer to the asymptotic case where we let the block length tend to infinity. The simplicity of the bounds derived in this paper stems from the fact that they are easily evaluated and are expressed in terms of some basic parameters of the ensemble which include the fraction of degree-$2$ variable nodes, the target bit erasure probability, and the gap between the channel capacity and the design rate of the ensemble. This paper demonstrates that the number of iterations which is required for successful message-passing decoding scales at least like the inverse of the gap (in rate) to capacity, provided that the fraction of degree-$2$ variable nodes of these turbo-like ensembles does not vanish (hence, the number of iterations becomes unbounded as the gap to capacity vanishes).
机译：本文提供了一些迭代次数的简单下限，这是成功地对一些重要的基于图形的代码集合（包括低密度奇偶校验（LDPC）代码和重复累积代码的变体）进行消息传递解码所需的迭代次数下限。 。假定代码集合的传输发生在二进制擦除通道上，并且边界是指渐近的情况，在此情况下，我们让块长度趋于无穷大。本文得出的边界之所以简单，是因为它们易于评估，并以整体的一些基本参数表示，包括度2元变量节点的分数，目标位擦除概率以及通道容量与整体设计速率之间的差距。本文证明，成功传递消息的解码所需要的迭代次数至少会像间隔（速率）与容量的倒数一样缩放，前提是这些turbo状集合的度$ 2 $可变节点的分数不会消失（因此，随着容量差距消失，迭代次数将变得不受限制）。

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