M. Sudan. Maximum likelihood decoding of reed solomon codes. In 37th Annual Symposium on Foundations of Computer Science, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....CS95, CS97, CNS96, Mel96] will note that the spirit of this idea is the same as in the sparse set results; our new result appears to require stronger techniques, though. The technique of [ALRS92] is quite powerful, and has already found some beautiful applications to decoding Reed Solomon codes [Sud96] and to the design of certain lowdegree tests [AS97] We believe that this technique could be a powerful tool in addressing other complexity theory questions as well. It is also amusing to note that our proof relates seemingly unconnected concepts such as p selectivity and bivariate polynomial ....

....shortlisting process, we have, for every u 2 F q , a list S u of at most q 1=3 elements that is guaranteed to contain the value of P a (u) The remaining task is to reconstruct the coefficients of P a , given these lists. For this, we will appeal to an algorithm of Ar et al. ALRS92] see also [Sud96] For the sake of completeness, and since the scenarios described in these papers do not exactly address our set up, we will describe the solution of this problem. We note, however, that all the necessary innovation and technology in the solution is present in [ALRS92] For simplicity of ....

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M. Sudan. Maximum likelihood decoding of Reed-Solomon codes. In Proc. 37th Annual IEEE Symposium on Foundations of Computer Science, pages 164--172, 1996.

....erasure codes , where one can recover the message word from a very small (asymptotically vanishing) fraction of the bits of the code word. An especially crucial component in all of our constructions is the recent surprisingly powerful polynomial reconstruction algorithm due to Madhu Sudan [Sud96] building on earlier work by Ar et al. ALRS92] This elegant algorithm has already found many complexity applications [AS97, Siv98, CPS98] Our bit level erasure code constructions based on this algorithm may also find other applications in complexity theory. 0.4 Extensions We generalize the ....

....that we use for the proof system constructions. All the codes we construct are compositions of the well known Reed Solomon code (or polynomial codes) with appropriately designed inner codes. We will rely heavily on the powerful decoder for Reed Solomon codes, developed by Madhu Sudan [Sud96] who builds on earlier work by Ar et al. ALRS92] Theorem 1 ( Sud96] Let F be a finite field. Given L distinct pairs (u i ; v i ) 2 F Theta F for i = 1; L, in time polynomial in L, d, and jFj, one can produce a list of at most p L=d polynomials over F that contains every polynomial ....

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M. Sudan. Maximum likelihood decoding of Reed-Solomon codes. In Proc. 37th Annual IEEE Symposium on Foundations of Computer Science, pages 164--172, 1996.

....but one which requires an artificial restriction on the data. Goldreich, Rubinfeld, et al. 4] using quite different methods, derive strong reconstruction results which, however, work only if the data is presented as an oracle which one can query at whatever locations one pleases. Finally Sudan [7, 8] was able to enhance the methods of [1] removing the artificial restriction. For the case of univariate polynomials, Sudan gives a surprisingly strong result [7, Theorem 5] from an arbitrary collection of n distinct data points, he efficiently reconstructs all degree ( d) polynomials that are ....

....solution set by throwing out any f which has total degree d or is consistent with t data points. Our bound on t, regarded as a function of n, is O(n k k 1 ) For k 2, this is superior to 3 the O(n 1 Gamma 1 k(k 1) bound of Theorem 1.1. We also require a lower bound on jF j (as do [7] and related papers) This bound will be further discussed in x6.2. Algorithm 2.1 is essentially the same as Sudan s Multivariate Algorithm [7, p. 169] We observe that the algorithm s solution set will contain at most L=d k 1 e k 1 q n=d k polynomials, so is suitably small. The ....

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M. Sudan, Maximum likelihood decoding of Reed Solomon codes, Proc. 37th IEEE FOCS, pp. 164-- 172, 1996.

....of integers. These results are a simultaneous improvement of the results in [GS92,FL92] as well as in [CH91] except replacing probabilistic for deterministic polynomial time algorithm in the latter) We achieve this improvement by applying an improved Reed Solomon decoder due to Madhu Sudan [Sud96], building on earlier work by Ar et al. ALRS92] Our proof uses a family of intermediate matrices whose entries are univariate polynomials in an indeterminate x; this definition unifies two ideas used in previous papers, and makes the proof simple and completely self contained. 2 Hardness over Z ....

....with high probability, we have a list f(x j ; y j )g of at most L pairs (for some L which is polynomially bounded in n) such that the graph of the polynomial per(D(x) of degree at most d = n 2 , intersects the list on more than p 2Ld places. Given such a list, the following lemma of Sudan [Sud96], building on earlier work by [ALRS92] shows how one can construct all the polynomials f whose graphs intersect the list on more than p 2Ld places. We note that Sudan s procedure is based on bivariate polynomial factoring. Therefore, it can be implemented in randomized polynomial time in L and ....

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M. Sudan. Maximum likelihood decoding of Reed-Solomon codes. In Proceedings of the 37th FOCS, pages 164--172, 1996.

....that could correct (d Gamma 1) 2 1 errors in time O(n 2 ) Dumer [6] developed an algorithm that could correct d=2 O(log n) errors in time O(n 2 ) Sidel nikov [35] gave another algorithm based on computing zeros of multivariate polynomials. A different line of thought was pursued by Sudan [40, 41], who, extending results of Ar et al. 1] investigated alternative decoding algorithms for Reed Solomon codes. By generalizing a decoding algorithm of Welch and Berlekamp [46, 2] he derived the surprising result that an [n; k] q ReedSolomon code is an [n; k; e; b] q code such that e is ....

M. Sudan. Maximum likelihood decoding of Reed-Solomon codes. In Proc. 37th FOCS, pages 164--172, 1996.

....is unclear when ffi 1=2, since as many as O(1=ffi) polynomials could have agreement ffi with the program. Two notions of correction are possible, as noted in [1] The weaker one is that for each input, the corrector outputs O(1=ffi) values, one of which is correct. Such a corrector is known [32]. The stronger notion is that the corrector create O( 1 ffi ) programs (polynomials) such that w.h.p. one of them is correct. Finding such a corrector was an open problem. Our analysis leads to such a corrector. Details of the proof are omitted from this abstract, but they are obvious from ....

....(Theorems 3 and 4) in Section 2. We prove the theorem in Section 3. This proof resembles proofs of earlier results [29, 5, 3, 17] in that it has two parts. First in Section 3. 1 we prove the theorem when m is constant (specifically, m = 2; 3) this uses algebraic arguments inspired by Sudan s [32] work on reconstructing polynomials from very noisy data and Kaltofen s work on Effective Hilbert Irreducibility [20, 21, 22] Then in Section 3.2 we bootstrap to allow larger m. This part uses probabilistic arguments and relies upon the cases m = 2; 3 (including Theorems 3 and 4 for the cases ....

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M. SUDAN. Maximum likelihood decoding of Reed Solomon codes. IEEE FOCS 1996.

....clear when ffi 1=2, since as many as O(1=ffi) polynomials could have agreement ffi with the program. Two notions of correction are possible, as noted in [ALRS92] The weaker one is that for each input, the corrector outputs O(1=ffi) values, one of which is correct. Such a corrector is given in [Su96]. The stronger notion is that the corrector creates O( 1 ffi ) programs (polynomials) such that w.h.p. one of them is correct. Finding such a corrector was an open problem. We provide such a corrector in Section 4.2. Past work. The first construction of a nontrivial constant prover 1 round ....

....6 and 7) in Section 2. We prove the theorem in Section 3. This proof resembles proofs of earlier results [RS93, ALMSS92, A94, FS95] in that it has two parts. First in Section 3. 1 we prove the theorem when m is constant (specifically, m = 2; 3) this uses algebraic arguments inspired by Sudan s [Su96] work on reconstructing polynomials from very noisy data and Kaltofen s work on Effective Hilbert Irreducibility [K85, K95] Then in Section 3.3 we bootstrap to allow larger m. This part uses probabilistic arguments and relies upon the cases m = 2; 3 (including Theorems 6 and 7 for the cases ....

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M. Sudan. Maximum likelihood decoding of Reed Solomon codes. Proceedings of the 37th Symposium on Foundations of Computer Science, IEEE, 1996.

....not clear when ffi 1=2, since as many as O(1=ffi) polynomials could have agreement ffi with the program. Two notions of correction are possible, as noted in [ALRS92] The weaker one is that for each input, the corrector outputs O(1=ffi) values, one of which is correct. Such a corrector is known [Su96]. The stronger notion is that the corrector create O( 1 ffi ) programs (polynomials) such that w.h.p. one of them is correct. Finding such a corrector was an open problem. Our analysis leads to such a corrector. We omit details from this abstract, but they are obvious from reading our proofs ....

....3 and 4) in Section 2. We prove the theorem in Section 3. This proof resembles proofs of earlier results [RS93, ALMSS92, A94, FS95] in that it has two parts. First in Section 3. 1 we prove the theorem when m is constant (specifically, m = 2; 3) this uses algebraic arguments inspired by Sudan s [Su96] work on reconstructing polynomials from very noisy data and Kaltofen s work on Effective Hilbert Irreducibility [K85, K95] Then in Section 3.3 we bootstrap to allow larger m. This part uses probabilistic arguments and relies upon the cases m = 2; 3 (including Theorems 3 and 4 for the cases ....

[Article contains additional citation context not shown here]

M. Sudan. Maximum likelihood decoding of Reed Solomon codes. Proceedings of the 37th Symposium on Foundations of Computer Science, IEEE, 1996.

....a black box over F on ffl fraction of the inputs, provided ffl 2 q d=jF j. Their algorithm runs in time O( n; 1 ffl ) poly(d) which is exponential in d. Their algorithm generalizes the earlier mentioned solution of Goldreich and Levin [15] For the case of univariate polynomials, Sudan [35], has given a polynomial time algorithm which can find all degree d polynomials agreeing with a black box on ffl fraction of the domain, provided ffl 2 q d=jF j. The main contribution in [35] is a simple observation which shows that m input output pairs from any black box can be thought of as ....

....earlier mentioned solution of Goldreich and Levin [15] For the case of univariate polynomials, Sudan [35] has given a polynomial time algorithm which can find all degree d polynomials agreeing with a black box on ffl fraction of the domain, provided ffl 2 q d=jF j. The main contribution in [35] is a simple observation which shows that m input output pairs from any black box can be thought of as the output of a (1; O( p m) algebraic black box. Using this observation, Lemma 18 of this paper is applied to reconstruct all polynomials of low degree which describe the black box on ....

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M. Sudan. Maximum likelihood decoding of Reed Solomon codes. Proc. 37th IEEE Symposium on Foundations of Computer Science, 1996.

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M. Sudan. Maximum likelihood decoding of reed solomon codes. In 37th Annual Symposium on Foundations of Computer Science, 1996.

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