# Introduction to coding theory (CMU, Spr 2010)

## April 24, 2010

### Lecture 24 summary

Filed under: Lecture summary — Venkat Guruswami @ 10:08 pm

We discussed the message passing algorithm for decoding LDPC codes based on $(d_v,d_c)$-regular graphs on the binary erasure channel, and derived an expression for threshold erasure probability for which the algorithm guarantees vanishing bit error probability. We then turned to the binary symmetric channel, and discussed Gallager’s “Algorithm A” and derived the recurrence equation for the decay of the bit error probability. We briefly discussed Gallager’s “Algorithm B” as well, where a variable node flips its value if more than a certain cut-off number (typically majority after a few iterations) of its neighboring check nodes suggest that the node flips its value. We mentioned the values of the threshold crossover probability for some small values of $d_v$ and $d_c$.

During lecture, the question of the speed of convergence of the bit error probability (BER) to zero was asked. The answer I guessed turns out to be correct: if we run the algorithm for $\Omega(\log n)$ iterations which is smaller than the girth of the graph, for Algorithm A the BER is at most $1/n^{\beta}$ for some $\beta > 0$, and for Algorithm B for $d_v > 3$ with an optimized cut-off for flipping, the BER is at most $2^{-n^{\gamma}}$ for some $\gamma > 0$.

We do not plan to have notes for this segment of the course.  I can, however, point you to an introductory survey I wrote (upon which the lectures are loosely based), or Gallager’s remarkable Ph.D. thesis which can be downloaded here (the decoding algorithms we covered are discussed in Chapter 4). A thorough treatment of the latest developments in the subject of iterative and belief propagation decoding algorithms can be found in Richardson and Urbanke’s comprehensive book Modern Coding Theory.