Jehoshua Bruck and Moni Naor. The hardness of decoding linear codes with preprocessing. IEEE Transactions on Information Theory, 36(2), March 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in Hamming distance to the target v. The NCP is also known to be hard to approximate. Namely, Arora et al. 2] show that approximating NCP to within 2 log (1 ) for any 0 is hard under the assumption that NP QP . We also de ne NCPP as the preprocessing variant of NCP. Bruck and Naor [6] show that the NCPP problem is NP hard to solve exactly. This was later extended by Micciancio [13] to the CVPP. However, both results apply only to the exact version of the problems and as noted in [13] it is not clear how to extend them to hardness of approximation. The rst inapproximability ....

J. Bruck and M. Naor. The hardness of decoding linear codes with preprocessing. IEEE Trans. Inform. Theory, 36(2):381-385, 1990.

....factors. For NEAREST CODEWORD and NV p for all p 1, we we can prove hardness up to a factor 2 instead of 2 . Also, in our reductions the number of variables, dimensions, input size etc. are polynomially related, so n could be any of these. Previous or independent work. Bruck and Naor ( BN] have shown the hardness of approximating the NEAREST CODEWORD problem to within some 1 ffl factor. Amaldi and Kann [AK93] have independently obtained results similar to ours for MAX SATISFY and MIN UNSATISFY. 2 Organization of the Paper Although the various problems mentioned in the ....

J. Bruck and M. Naor. The hardness of decoding linear codes with preprocessing. IEEE Transactions on Inform. Theory 1990 381-385.

....problem; 2) The code is speci ed as part of the input, rather than being a well known one which is more standard. It would be nice to know which of the two causes is responsible for the hardness, since for all well known codes, the error correction problem seems to well solved. Bruck and Naor [6] present a code (not well known, but nevertheless easily presented) for which they show that the existence of small size maximum likelihood decoding circuits would imply the collapse of the polynomial hierarchy (using a result of Karp and Lipton s [14] However the codes presented by Bruck and ....

J. Bruck and M. Naor. The hardness of decoding linear codes with preprocessing. IEEE Transactions on Information Theory, pp. 381-385, 1990.

....complexity assumptions. In particular, we show that if the closest vector problem with preprocessing can be solved in polynomial time, then NP is contained in P=poly and the polynomial hierarchy collapses (see [18] Our result is analogous to similar results for the nearest codeword problem [19] and the subset sum problem [20] and is based on a new proof of the NP hardness of the closest vector problem. A related result is presented in [17] where it is proved that for recursive cube search (RCS) algorithms (including the algorithm of Kannan [16] the complexity of decoding any ....

....a i x i = xn 1)s. Obviously, there is no guarantee that the x i are always 0 or 1, or that xn 1 = 1, and the LagariasOdlyzko algorithm succeeds only with high probability when the density of the lattice is suciently small. The closest vector problem (as well as the decoding problem studied in [19]) are known to be NP hard not only to solve exactly, but also when one seeks only an approximate solution [25] 26] The arguments presented in this paper for lattices and in [19] 17] for the codes do not seem to extend to approximation versions of the problems. We leave as an open problem to ....

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Jehoshua Bruck and Moni Naor, \The hardness of decoding linear codes with preprocessing," IEEE Transactions on Information Theory, vol. 36, no. 2, pp. 381-385, Mar. 1990.

....to code distance) Thus the hardness of the NCP may come from one of two factors: 1) The problem attempts to decode every linear code and (2) The problem attempts to recover from too many errors. Both issues have been raised in the literature [17] but only the former has seen some progress [6]. One problem that has been de ned to study the latter phenomenon is the Bounded distance decoding problem (BDD, see [17] This is a special case of the NCP where the error weight is guaranteed (or promised ) to be less than d(A) 2: This case is motivated by the fact that within such a ....

....the input. Thus it is still conceivable that for every error correcting code, there exists a fast algorithm to correct errors (say up to the distance of the code) however, this algorithm may be hard to nd (given a description of the code) A result along the lines of the result of Bruck and Naor [6], showing the hardness of relatively nearby codeword problem even with preprocessing, would be desirable to x this gap in our knowledge. We are however unable to extend our techniques (or those of [6] and [13] to address this problem. ....

[Article contains additional citation context not shown here]

J. Bruck, M. Naor, \The Hardness of Decoding Linear Codes with Preprocessing," IEEE Transactions on Information Theory, Vol. IT-36, n. 2, March 1990, pp. 381-385.

....distance of the code. Thus the hardness of the NCP may come from one of two factors: 1) The problem attempts to decode every linear code and (2) The problem attempts to recover from too many errors. Both issues have been raised in the literature [15] but only the former has seen some progress [6]. One problem that has been defined to study the latter phenomenon is the Bounded distance decoding problem (BDD, see [15] This is a special case of the NCP where the error is guaranteed (or promised ) to be less than half the minimum distance of the code. This case is motivated by the fact ....

J. Bruck, M. Naor, "The Hardness of Decoding Linear Codes with Preprocessing," IEEE Transactions on Information Theory, Vol. IT-36, n. 2, March 1990, pp. 381-- 385.

....the same code may not yield an equally efficient algorithm for decoding. In fact this non trivial feature laid the foundation for a public key cryptosystem proposed by McEliece [11] On the other hand some error correcting codes are inherently hard to decode. This was established by Bruck and Naor [3] who proved that there are codes which are hard to decode even with unlimited preprocessing time and hence independent of representation. Our result shows that algebraic geometric codes (henceforth AG codes) do not belong to this class, i.e. they admit a polynomial size representation given which ....

J. Bruck and M. Naor. The hardness of decoding linear codes with preprocessing. IEEE Trans. on Information Theory, Vol. 36, No. 2, March 1990.

....problem; 2) The code is speci ed as part of the input, rather than being a well known one which is more standard. It would be nice to know which of the two causes is responsible for the hardness, since for all well known codes, the error correction problem seems to well solved. Bruck and Naor [6] present a code (not well known, but nevertheless easily presented) for which they show that the existence of small size maximum likelihood decoding circuits would imply the collapse of the polynomial hierarchy (using a result of Karp and Lipton s [13] However the codes presented by Bruck and ....

J. Bruck and M. Naor. The hardness of decoding linear codes with preprocessing. IEEE Trans. Inform. Theory, pp. 381-385, 1990.

....d(x; y) is the minimum possible. If s = Hy t , this is equivalent to finding the smallest set of columns of H that sums to s; hence the corresponding decision problem is precisely Maximum Likelihood Decoding. 6 While the complexity of maximum likelihood decoding has been thoroughly studied [ABSS93, Bar94, BMvT78, BN90, Ste93], almost nothing is presently known regarding the complexity of bounded distance decoding, even though most of the decoders used in practice are boundeddistance decoders. A decoder is said to be bounded distance if there exists a constant t 0 such that for all y 2 IF n 2 , the decoder always ....

J. Bruck and M.Naor, The hardness of decoding linear codes with preprocessing, IEEE Trans. Inform. Theory, 36, (1990), 381--385.

....distance of the code. Thus the hardness of the NCP may come from one of two factors: 1) The problem attempts to decode every linear code and (2) The problem attempts to recover from too many errors. Both issues have been raised in the literature [16] but only the former has seen some progress [6]. One problem that has been defined to study the latter phenomenon is the Bounded distance decoding problem (BDD, see [16] This is a special case of the NCP where the error is guaranteed (or promised ) to be less than half the minimum distance of the code. This case is motivated by the fact ....

....input. Thus it is still concievable that for every error correcting code, there exists a fast algorithm to correct errors (say up to the distance of the code) however, this algorithm may be hard to find (given a description of the code) A result along the lines of the result of Bruck and Naor [6], showing the hardness of relatively nearby codeword problem even with preprocessing, would be desirable to fix this gap in our knowledge. We are however unable to extend our techniques (or those of [6] to address this problem. ....

[Article contains additional citation context not shown here]

J. Bruck, M. Naor, "The Hardness of Decoding Linear Codes with Preprocessing," IEEE Transactions on Information Theory, Vol. IT-36, n. 2, March 1990, pp. 381--385.

....(2) The code is specified as part of the input, rather than being a well known one which is more standard. It would be nice to know which of the two causes is responsible for the hardness, since for all well known codes, the error correction problem seems to well solved. Bruck and Naor [6] present a code (not well known, but nevertheless easily presented) for which they show that the existence of small size maximum likelihood decoding circuits would imply the collapse of the polynomial hierarchy (using a result of Karp and Lipton s [14] However the codes presented by Bruck and ....

J. Bruck and M. Naor. The hardness of decoding linear codes with preprocessing. IEEE Transactions on Information Theory, pp. 381--385, 1990.

....algorithm for solving a problem in the worst case; however many NP complete problems can be attacked after a suitable preprocessing phase or can be suitably approximated or else can be efficiently solved on average. The first issue has been discussed in a related work of Bruck and Naor ([5]) where they present a linear code, such that even with a large amount of pre processing (based on the parity matrix only) it is still hard to produce a minimum weight word leading to this syndrome. Non approximability results for the minimum distance of a code appear in [1] As for the the ....

J. Bruck and M. Naor. The hardness of decoding linear codes with preprocessing, IEEE Trans. Inform. Theory, IT-36(2) (1980), 381--385.

.... and some have formally been proved NP complete: for example, Linear Decoding, the general problem of maximumlikelihood decoding an arbitrary binary linear block code used on a binary symmetric channel [9] This problem remains hard even when arbitrary preprocessing of a given code is allowed [10]. Similarly, the problems of finding a codeword with least weight not a multiple k, finding a maximum weight codeword, or finding a codeword whose weight is in a given range in a binary linear block code are NP complete [11] Even the problem of minimizing the vertex count in the trellis of an ....

J. Bruck and M. Naor, "The hardness of decoding linear codes with preprocessing," IEEE Trans. on Inform. Theory, vol. 36, pp. 381--385, March 1990.

....of K 1 can be polynomially transformed to an instance of P 9 in such a way that U has a matching if and only if the question of P 9 can be answered in the affirmative. That this implies our claim will then follow from a general structure theorem on nonuniform complexity, discussed for instance in [33]. Let W be a t set, N = t 3 , and let us number the triples u j 2 W 3 in a certain fixed way. Form a ( 3t 2N ) Theta 3N ) matrix H = 2 6 4 B I k I k I k I k 3 7 5 ; where B is a 3t Theta N incidence matrix of the set W 3 . Let s = 11 : 1; zU ; zU ) be a (3t 2N) vector, where ....

....Ntafos and Hakimi [124] finding a basis of minimal total weight, see Chickering et al. 40] Horton [85] and computing the covering radius, see Frank [66] Theorem 4. 4 is due to McLoughlin [122] The complexity of the Weight of error problem with preprocessing (P 9 ) was proved in Bruck and Naor [33]. Their reduction is from K 3 . The proof that we give, following Lobstein [24, pp. 121 123] has the advantage of being valid for any fixed code alphabet. The nonapproximability theorem (Theorem 4.6) and the surrounding discussion is from Arora et al. 11] Stern [151] Stern [151] constructs a ....

J. Bruck and M. Naor, "The hardness of decoding linear codes with preprocessing, " IEEE Trans. Inform. Theory, IT-36 (2) (1990), 381--385.

....NV p for all p 1, we we can prove hardness up to a factor 2 log 1 Gammaffl n instead of 2 log 0:5 Gammaffl n . Also, in our reductions the number of variables, dimensions, input size etc. are polynomially related, so n could be any of these. Previous or independent work. Bruck and Naor ( BN] have shown the hardness of approximating the NEAREST CODEWORD problem to within some 1 ffl factor. Amaldi and Kann [AK93] have independently obtained results similar to ours for MAX SATISFY and MIN UNSATISFY. 2 Organization of the Paper Although the various problems mentioned in the ....

J. Bruck and M. Naor. The hardness of decoding linear codes with preprocessing. IEEE Transactions on Inform. Theory 1990 381-385.

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Jehoshua Bruck and Moni Naor. The hardness of decoding linear codes with preprocessing. IEEE Transactions on Information Theory, 36(2), March 1990.

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