Posted by: atri | December 5, 2007

Lecture 41:Parvaresh Vardy Decoder

In today’s lecture, we saw how to decode folded RS codes of rate R up to 1-\sqrt[m+1]{(mR)^m} fraction of errors. For suitable choice of parameters, one can correct 1-\epsilon fraction of errors with rate \Omega(\epsilon/\log(1/\epsilon)) (I think the stated an incorrect bound on the rate in class). We still need to prove the following lemmas

  • There exists an irreducible polynomial E(X) of degree q-1 such that for any polynomial f(X) of degree at most q-2, the following is true: f(X)^q\equiv f(\gamma X)\mod(E(X)), where \gamma is the generator of the field \mathbb{F}_q.
  • Given an polynomial f(X) that needs to be output in the second step of the list decoding algorithm, we have T(f(X),f(\gamma X))\equiv 0, where f(X) and f(\gamma X) are thought of as elements of \mathbb{F}_q[X]/E(X)\equiv \mathbb{F}_{q^{q-1}}. Recall that T(Y,Z) was defined as T_0(Y,Z)\mod(E(X)), where T_0(Y,Z) (with coefficients from \mathbb{F}_q[X]) is just Q_0(X,Y,Z). Finally, recall Q_0(X,Y,Z) is the largest factor of Q(X,Y,Z) from step 1 of the algorithm that is not divisible by E(X).

In the next (and last!) lecture, we will quickly prove the above two lemmas. Then we will see how a small change to the algorithm allows us to decode from 1-R-\epsilon fraction of errors in polynomial time (for constant \epsilon>0). If we have more time, we will either briefly cover list decoding of binary codes and/or topics that we did not cover in any detail in this class.

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