Posted by: atri | November 16, 2007

## Lecture 35: Threshold Computation

In today’s lecture we computed the threshold $\alpha^*$ such that the iterative message passing algorithm has exponentially small decoding error probability for $BEC_{\alpha}$ for every $\alpha<\alpha^*$. Michel asked what happens when $\alpha>\alpha^*$. In this case, it can be shown that even in the limit (that is, when the number of rounds is infinite) the probability of an erasure being passed on an edge is strictly bounded away from zero. In other words, the decoding algorithm is going to have a constant decoding error probability irrespective of the number of iterations. See Guruswami‘s survey for a formal proof.

Next lecture, we will see how the decoder can be implemented in linear time and (at a very very high level) why irregular LDPC codes can have a greater threshold than regular LDPC codes (such codes achieve the capacity of $BEC_{\alpha}$). We will then move on to algorithms for list decoding.

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