On the Performance of Multivariate Interpolation Decoding of Reed-Solomon Codes Farzad Parvaresh University of California San Diego La Jolla, CA 92093, U.S.A. fparvaresh@ucsd.edu Mohammad H. Taghavi University of California San Diego La Jolla, CA 92093, U.S.A. mtaghavi@ucsd.edu Alexander Vardy University of California San Diego La Jolla, CA 92093, U.S.A. vardy@kilimanjaro.ucsd.edu Abstract— The multivariate interpolation decoding (MID) al- gorithm for certain Reed-Solomon codes was recently introduced by Parvaresh and Vardy. The MID algorithm attempts to list-de- code up to nτ MID = n ( 1 - R M/( 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 τ GS = 1- √ 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 nτ 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 nτ 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 de- tailed analysis of the probability of failure in the MID algorithm for the special case where M = 2 and the interpolation multipli- city is m = 1. In this case, the MID algorithm attempts to correct up to nτ 2,1 errors, where τ 2,1 = 1 - 3 √ 6R 2 . We consider the sit- uation where symbol values received from the channel at the er- roneous positions are distributed uniformly at random (a version of the q-ary symmetric channel). We show that, with high proba- bility, 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 τ 2,1 - O(R 5/3 ), then the probabil- ity of failure in the MID algorithm is at most n -Ω(n) . Thus the probability of failure is, indeed, negligible for large n in this case. I. I NTRODUCTION Reed-Solomon codes are ubiquitous, with applications rang- ing from magnetic recording to deep-space communications. The classical decoding algorithms (employed in all these ap- plications today) correct up to 1 / 2 n(1−R) errors in a Reed- Solomon code of length n and rate R. In two breakthrough papers, Sudan [9] and Guruswami-Sudan [6] showed that we can do much better: Reed-Solomon codes could be used to cor- rect up to n(1− √ R) errors, if the decoder is allowed to output a small list of codewords. As it turns out, in practice, the prob- ability that the Guruswami-Sudan decoder actually produces more than one codeword is usually negligible (cf. [8]). More recently, several papers [1–3,5,10–12] showed that the Guruswami-Sudan decoding radius τ GS = 1− √ R could be sub- stantially improved upon, under certain conditions. All these papers consider the following scenario: M codewords of an (n, k, d) Reed-Solomon code C over F q are decoded together and the errors are assumed to be synchronized — that is, the error positions are the same for all the M codewords. Alter- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rate of the code Fraction of errors corrected Berlekamp-Massey (1/2)(1-R) CS 1-R-R 2/3 BKY (2/3)(1-R) MID 1-R 2/3 Figure 1. Error-correction radii of BKY, CS, and MID algorithms natively, one can think of the entire M × n array, having the M codewords of C as its rows, as a single codeword of an (n, k, d) code C over the field of order Q = q M . Then, if a codeword of C is corrupted by some t errors, these errors are necessarily synchronized with respect to the constituent codewords of C. Curiously, the code C is actually a Reed-Solomon code over F Q (as shown in [12]). Polynomial-time decoding algorithms for this framework have been recently developed by Bleichen- bacher, Kiayias, and Yung [1], by Coppersmith and Sudan [3], and by Parvaresh and Vardy [10,12]. The three algorithms are based upon very different ideas, but have several common fea- tures. First, all the three attempt to decode significantly beyond the the Guruswami-Sudan decoding radius τ GS = 1− √ R. The error-correction radii of the three algorithms are given by τ BKY = M M+1 ( 1−R ) , τ CS = 1− R − R M M+1 , τ MID = 1 − R M M+1 respectively. These are plotted in Figure1 for the special case M = 2. Second, in all the three cases, successful decoding up to this radius is not guaranteed: there are certain specific error patterns of less than nτ BKY (or nτ CS , or nτ MID , respectively) er- rors which cause a decoding failure. The probability of such decoding failure for the BKY and CS decoders is analyzed in great detail in [1,2] and in [3], respectively. It would be nice to have a similar probabilistic analysis for the multivari- ate interpolation decoder (MID) of Parvaresh and Vardy [10]. Unfortunately, this task appears to be extremely difficult; the present work can be regarded as the first step toward this goal. As suggested by the referees, it might be worth indicating here some of the advantages of the multivariate interpolation approach of [10,12] over the alternatives. First as can be seen ISIT 2006, Seattle, USA, July 9  14, 2006 2027 1424405041/06/$20.00 ©2006 IEEE