On the performance of multivariate interpolation decoding of Reed-Solomon codes

The multivariate interpolation decoding (MID) algorithm for certain Reed-Solomon codes was recently introduced by Parvaresh and Vardy. The MID algorithm attempts to list-decode up to n tau(MID) = n(1 - RM/(M+1)) errors, in a Reed-Solomon code of length n and rate R, using (M+1)-variate polynomial interpolation. This improves on the Guruswami-Sudan decoding radius of tau(GS) = 1 - root R by a large margin, especially for high-rate codes. The problem is that successful decoding is not guaranteed: there are certain patterns of less than n tau(MID) errors which the MID algorithm fails to decode. Nevertheless, simulations show that the actual performance of the MID decoder is very close to what one would expect if all patterns of up to n tau(MID) errors were corrected. On the other hand, analysis of the failure probability for the MID algorithm is extremely difficult, and there were no analytic results so far to confirm this empirically observed behavior. In this work, we provide such analytic results: we present a detailed analysis of the probability of failure in the MID algorithm for the special case where M = 2 and the interpolation multiplicity is m = 1. In this case, the MID algorithm attempts to correct up to n tau(2,1) errors, where tau(2,1) = 1 -(3)root 6R(2). We consider the situation where symbol values received from the channel at the erroneous positions are distributed uniformly at random (a version of the q-ary symmetric channel). We show that, with high probability, the performance of the MID algorithm is very close to the optimum in this case. Specifically, we prove that if the fraction of positions in error is at most tau(2,1) - O(R (5/3)), then the probability of failure in the MID algorithm is at most n(-Omega(n)). Thus the probability of failure is, indeed, negligible for large n in this case.