Deterministic versions of these algorithms takes polylog f, n steps over most fields. In this paper, it is proved under a well known cryptography hardness assumption that. The second possibility is a graphical representation. This is the problem tackled in this paper for the case of reed solomon codes. Pdf decoding of reed solomon codes beyond the error. In this paper, we prove that maximum likelihood decoding is np hard for the family of reed solomon codes. Maximumlikelihood decoding is one of the central problems in coding theory. We moreover show that maximum likelihood decoding of reed. In particular this opens the possibility of syndrome decoding.
It has been knownfor over25 years that maximumlikelihooddecoding of general linear codes is nphard. Iterative algebraic softdecision list decoding of reedsolomon. In a previous paper 8, we showed that the received word u is a deep hole of the standard reedsolomon codes q1, k q if its lagrange interpolation polynomial is the sum of monomial of degree q2 and a polynomial of degree at most k1. Generalized reedsolomon codes the coe cients l i i are always nonzero and are often easy to compute. It remains an open problem to find polynomialtime decoding algorithms with near ml performance. Nevertheless, it was so far unknown whether maximumlikelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure.
Equation 4 illustrates that for the case of rs codes, correcting. Generalized reedsolomon codes michigan state university. For these codes, we show that the maximumlikelihood. Decoding of reed solomon codes beyond the errorcorrection bound madhu sudan abstract we present a randomized algorithm which takes as input ndistinct points fx i. Nevertheless, it was so far unknown whether maximumlikelihood decoding remains hard forany speci.
Likelihood decoding of reedsolomon code is npcomplete. Decoding of reed solomon codes beyond the errorcorrection. In fact by making the task enumerative rather than. Iterative decoding of convolutional codes applied to. Maximumlikelihood decoding of reedsolomon codes is. The maximum likelihood decoding problem is known to be np hard for general linear and reed solomon codes 1, 4. The maximumlikelihood decoding problem is known to be nphard for general linear and reedsolomon codes 1, 4. The reed solomon codes would have formed a good candidate to show the hardness of. Since that time theyve been applied in cdroms, wireless communications, space communications, dsl, dvd, and digital tv. For the complexity of maximum likelihood decoding of the primitive reedsolomon code, whose length is one less than the size of alphabet, the only known result states that it is at least as hard as. In this paper, we introduce the notion of acovered codes, that is, codes that can be. These codes have great power and utility, and are today found.
This is a difficult theoretical problem in general. Reedsolomon codes also play a fundamental role in modern. Lecture 8 1 overview 2 list decoding of reedsolomon codes. In this lecture we will cover list decoding of reedsolomon codes. Moreover, we show that the problem is nphard under quasipolynomialtime reductions for an error amount.
Pdf maximumlikelihood decoding of reedsolomon codes is np. In fact, the weaker problem of deciding deep holes for generalized reedsolomon codes is already conpcomplete, see 3. Maximumlikelihood decoding is one of the central algorithmic problems in codingtheory. On deep holes of standard reedsolomon codes springerlink. Determining deep holes is an important topic in decoding reedsolomon codes. Separate coding has the advantage that decoding is easy. Pdf new set of codes for the maximumlikelihood decoding. Pdf maximumlikelihood decoding of reedsolomon codes is. Guruswami and vardy 6 proved the problem to np hard over exponentially large fields. Software implementation of the reedsolomon encoder and decoder, and additionally parts of the. In this paper, we extend this result by giving a new class of deep holes of the. Abstractit has been proved that the maximum likelihood decoding problem of reedsolomon codes is nphard. Pdf the maximum likelihood decoding problem is known to be np hard for general linear and reed solomon codes.
Software implementation of the reedsolomon encoder and decoder, and additionally parts of. In this paper, we prove that maximumlikelihood decoding is nphard for the family of reedsolomon codes. New set of codes for the maximumlikelihood decoding problem. Separate coding has the advantage that decoding is.
Euclidean algorithm for bch codes guruswamisudan list decoding for reedsolomon codes probabilistic ordered statistics decoder. Reed solomon encoder and decoder rutgers university. These algorithms attempt to bridge the often signi cant gap between maximum likelihood ml. Complexity of decoding positiverate reedsolomon codes.
The maximum likelihood decoding of a generalized reedsolomon n, kq code is known to be np complete. Likelihood ml decoding, is a difficult problem concerning the complexity. New set of codes for the maximumlikelihood decoding. However, the length of the code in the proof is at most polylogarithmic in the size of the alphabet. Complexity analysis shows that the lccbr algorithm yields a lower complexity and latency, especially for high rate codes, which will be validated by the numerical results. Speci cally, for reedsolomon codes of length nand dimension k n, we show that it is nphard to decode more than n k c log n log log n errors with c0 an absolute constant. Decoding of reed solomon codes beyond the errorcorrection bound. Let q 2m and let f qx denote the ring of univariate polynomials over f q. Solomon code in a binary vanishing channel when combined with convolutional code.
May 04, 2004 in this paper, we prove that maximumlikelihood decoding is nphard for the family of reedsolomon codes. A reedsolomon code is a bch code over gfq of length n q 1, that is, m 1. Reedsolomon codes by bernard sklar introduction in 1960, irving reed and gus solomon published a paper in the journal of the society for industrial and applied mathematics 1. Soft decision decoding of reedsolomon product codes. One could then decode \mathbfy to this message px as the maximum likelihood choice. So the generator polynomial of a reedsolomon code of designed distance. Nevertheless, it was so far unknown whether maximum likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure.
Reedsolomon codes qi cheng and daqing wan abstractit has been proved that the maximum likelihood decoding problem of reedsolomon codes is nphard. For reedsolomon codes, characterization is neither trivial nor explicitly wellstudied, but, both in the original proof that maximum likelihood decoding is nphard 3 and the proof we present here, deep holes seem to be the reason that certain types of decoding are nphard. Nphardness of reedsolomon decoding and the prouhet. Maximumlikelihood decoding of reedsolomon codes is np. Reedsolomon code is also cyclic code, so decoding algorithms for cyclic codes can be used. As we have seen before, describing a linear code instead in terms of a check matrix can be fruitful. The reedsolomon code generator polynomial used was based off of the n255, k239 code. Wolf showed that soft ml decoding for any linear block code is possible using viterbi algorithm on trellises. Rs codes are seen as a special case of the larger class of bch codes but it was not until almost a decade later, by regarding them as cyclic bch codes.
Hocquenghem code and particular decoding algorithms for bch code can be used together for reedsolomon code. Reedsolomon codes are obtained by evaluating certain subspaces off qx in a set of points d x 1,x 2. Maximumlikelihood decoding of reedsolomon codes is nphard abstract. It has been known for over 25 years that maximumlikelihood decoding of general linear codes is nphard. Improvements on the johnson bound for reed solomon codes. Vardy, a maximumlikelihood decoding of reedsolomon codes is nphard.
Reedsolomon codes 1 introduction a reedsolomon rs code is an errorcorrecting code rst described in a paper by reed and solomon in 1960 9. Nevertheless, it was so far unknown whether maximumlikelihood decoding remains hard for any specific family of codes with. The holy grail in decoding rs codes would be to find the polynomial px whose rs encoding. Newsetofcodesforthemaximumlikelihood decodingproblem. In most testing the code was shortened to n32, k16 via code shortening populating the initial 23916 symbols with zeros. The maximumlikelihood decoding problem of reedsolomon codes is. The complexity of maximal likelihood decoding of the reed. There are wellknown polynomialtime algorithms that decode reedsolomon codes up to half their minimum distance 10, 18, 24, and also well beyond half the minimum distance 12, 21. It has been proved that the maximum likelihood decoding problem of reedsolomon codes is nphard. Complexity of decoding positiverate primitive reed solomon. For the complexity of maximum likelihood decoding of the primitive reedsolomon code, whose. Separate coding can achieve better characteristics than reed. Complexity of decoding positiverate primitive reedsolomon codes.
Low complexity soft decision decoding algorithms for reed. For a reed solomon code with parameters n block size, k message size, q symbol size in bits, we encode the message as a polynomial px, and then multiply with a code generator polynomial gx map the message vector x 1 x k to a polynomial px of degree decoding is one of the central algorithmic problems in coding theory. Index termsbasis reduction, lowcomplexity chase decoding, progressive decoding, reedsolomon codes. For standard reedsolomon code with d f q, the complexity of the maximum likelihood decoding is unknown to be npcomplete. Note that decoding reedsolomon codes beyond the johnson bound remains a fully open problem. Berlekamp, mceliece and van tilborg showed in 1 that the maximumlikelihood decoding is a nphard problem for general linear codes.
Maximumlikelihood decoding is one of the central algorithmic problems in coding theory. This paper described a new class of errorcorrecting codes that are now called reedsolomon rs codes. Citeseerx document details isaac councill, lee giles, pradeep teregowda. For a reed solomon code with parameters n block size, k message size, q symbol size in bits, we encode the message as a polynomial px, and then multiply with a code generator polynomial gx map the message vector x 1 x k to a polynomial px of degree decoding problem is known to be nphard for general linear and reedsolomon codes. It was shown by chengwan 9, 10 that decoding the standard reedsolomon code is at least as hard as the discrete logarithm problem in a large extension of the. In the much more interesting case of standard reedsolomon codes, it is unknown if decoding remains nphard. Guruswami and vardy later proved in 4 that this problem applied to the family of reedsolomon codes is also nphard. Jun 27, 2005 maximumlikelihood decoding of reedsolomon codes is nphard abstract. Lowcomplexity chase decoding of reedsolomon codes using module. Reedsolomon codes reed and solomon, 1960 are a special class of bch codes. In this paper, we prove that maximumlikelihood decoding is nphardfor the family of reedsolomon codes. Much of the recent research about reedsolomon codes has come from a new approach to decoding reedsolomon codes, developed initially by sudan 5.
There are wellknown polynomialtime algorithms that decode reedsolomon codes up to half their minimum distance 3, 10, 18, and also well beyond half the minimum distance 12, 21. Maximumlikelihood decoding of reedsolomon codes is nphard. However, the codes presented by bruck and naor do not have a large distance. We moreover show that maximumlikelihood decoding of reedsolomon codes remains hard even with unlimited preprocessing, thereby strengthening a result of bruck and naor. The latter task is called nearestcodeword decoding or maximum likelihood decoding mld. A listdecoding approach to lowcomplexity soft maximum. Like all linear block codes they can be described via matrices. Complexity of decoding positiverate primitive reed. The only known result 4 in this direction states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero.
The complexity of maximum likelihood decoding of the reedsolomon codes q. Rs codes are seen as a special case of the larger class of bch codes but it was not until almost a decade later, by regarding them as cyclic bch codes, that. In 1960 2 peterson presented a method for decoding bch codes and in 1961 3 gorenstein and zierler tailored petersons method to reed. Maximumlikelihood decoding of reedsolomon codes is nphard eaiae ia. In this paper, we prove that maximumlikelihood decoding is. On the error distance of extended reedsolomon codes.
The reed solomon codes would have formed a good candidate to show the hardness of this problem, except that it is not as hard to solve. Iterative decoding of convolutional codes applied to separate. For the complexity of maximum likelihood decoding of the primitive reedsolomon code, whose length is one. To the best of our knowledge, this is the rst e cient i. In this paper, we introduce the notion of acovered codes, that is, codes that can be decoded through a polynomial time algorithm a whose decoding bound is beyond the covering radius. Home conferences soda proceedings soda 05 maximumlikelihood decoding of reedsolomon codes is nphard. Nov 12, 2010 the maximumlikelihood decoding problem is known to be nphard for general linear and reedsolomon codes. These reedsolomon product codes rspc are used in the encoding of data for dvds. Separately, maximumlikelihood ml decoding can be performed by the viterbi algorithm 6 on any trellis representation of the cyclic code. Decoding of reedsolomon codes is a wellstudied problem with a long history. It still remains open if the maximum likelihood decoding problem is hard for any constant distance code. Rs encoding data is relatively straightforward, but decoding is time. Nphardness of reedsolomon decoding and the prouhettarry.
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