Introduction to coding theory (CMU, Spr 2010)

April 28, 2010

Lecture 25 summary

Filed under: Lecture summary — Venkat Guruswami @ 8:51 pm

We discussed irregular LDPC codes, and characterized their rate and erasure correction capability (via the message passing algorithm discussed in the previous lecture) in terms of the degree distribution of the edges. Specifically, let \lambda_i (resp. \rho_i) is the fraction of edges incident on degree i variable (resp. check) nodes, an define the generating functions \lambda(z) = \sum_{i=1}^{d_v^{\max}} \lambda_i z^{i-1} and \rho(z) = \sum_{i=1}^{d_c^{\max}} \rho_i z^{i-1}. Then the rate of the LDPC code is given by

\displaystyle 1 -\frac{\int_0^1 \rho(z) \ dz}{\int_0^1 \lambda(z) \ dz} \ .

Also if \alpha \lambda(1-\rho(1-x)) \le x for every x, 0 \le x \le 1, and some constant \alpha > 0, we argued why the message passing algorithm succeeds with high probability on \mathrm{BEC}_\alpha' for any constant \alpha' < \alpha.

We then argued how the distributions

\displaystyle \lambda(z) = \frac{1}{H(D-1)} \sum_{i=1}^{D-1} \frac{z^i}{i}

and

\displaystyle \rho(z) = \exp \left( \frac{H(D-1)}{\alpha} (z-1) \right)

(perhaps truncated to a finite series) enables achieving capacity of \mathrm{BEC}_{\alpha'} — we can achieve a rate 1-\alpha'-\epsilon with decoding complexity O(n \log (1/\epsilon) (since the average variable node degree is \approx H(D-1)).

This result is from the paper Efficient erasure correcting codes. Further details, including extensions to BSC and AWGN channels, and the martingale argument for the concentration of the performance around that of the average code in the ensemble, can be found in the paper The capacity of low-density parity-check codes under message-passing decoding.

The last quarter of the lecture was devoted a recap of the main topics covered in the course.

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