I. Dumer, D. Micciancio, and M. Sudan. Hardness of approximating the minimum distance of a linear code. In IEEE Transaction on Information Theory, 49(1), 2003.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....result since both were shown for the decision problem. In the above we considered the preprocessing variant of NCP as a way to nd whether the hardness of NCP is a result of the algorithm required to work with any linear code. Another reason for the hardness of NCP was raised in the literature ([8]) It might be hard to nd the closest word to v because v is very far from any codeword. In other words, we are trying to recover from too many errors. The Relatively Near Codeword with parameter 0 (RNC ) is de ned as follows. Given a generating matrix C 2 F , a target vector v 2 F ....

....t from v. The algorithm is expected to work only when such a codeword exists. The problem becomes easier as decreases and many classical error correcting algorithms work in polynomial time when 2 . For = 2 we are guaranteed to have exactly one codeword within distance t from v. In [8] it is shown that RNC for any 2 is hard to approximate to within any constant factor. In this paper we consider RNCP , the preprocessing variant of the problem. The problem was mentioned in [10] but the authors were not able to provide any inapproximability result. The RNCP is ....

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I. Dumer, D. Micciancio, and M. Sudan. Hardness of approximating the minimum distance of a linear code. In Proc. 40th IEEE Symp. on Foundations of Computer Science, pages 475-484, 1999.

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Ilya Dumer, Daniele Micciancio, and Madhu Sudan. Hardness of approximating the minimum distance of a linear code. Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, pages 475-484, 1999.

....is acceptable for the applications we describe) The following lemma proves that with high probability many polynomials degree less than k agree with the target set R in t places, i.e. that there are many spurious polynomials. This lemma and its proof are based on similar results of Dumer et al. [9]. Recall that the locking algorithm Lock picks t points according to a given p of degree less than k and r t random points (x i , y i ) in and outputs this set in random order as a vault hiding p (i.e. #) Recall that q denotes the cardinality of . The following lemma is parameterized ....

I. Dumer, D. Micciancio, and M. Sudan. Hardness of approximating the minimum distance of a linear code. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 475--484, 1999.

....not apply for codes over any fixed size alphabet, and in particular for binary codes. The quantity L ( is even less well understood. When is either very large (of the form 1=2 o(1) or very small (of the form o(1) there is some evidence confirming this bound. In particular, Dumer et al. [6] construct a family of linear codes C, for any 0, for which (n) n ( 2 ) which matches the conjecture above reasonably closely. We give a simple probabilistic argument to show the following: Theorem 10: For every 0, there exists an infinite family of binary codes C and a ....

I. Dumer, D. Micciancio and M. Sudan. Hardness of approximating the minimum distance of a linear code. Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), New York, NY, October 1999, pp. 475-484.

....apply for codes over any fixed size alphabet, and in particular for binary codes. The quantity L poly ( is even less well understood. When is either very large (of the form 1=2 o(1) or very small (of the form o(1) there is some evidence confirming this bound. In particular, Dumer et al. [6] construct a family of linear codes C, for any 0, for which (n) n 1 and L poly ( 2 ) which matches the conjecture above reasonably closely. We give a simple probabilistic argument to show the following: Theorem 10: For every 0, there exists an infinite family of binary codes ....

I. Dumer, D. Micciancio and M. Sudan. Hardness of approximating the minimum distance of a linear code. Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), New York, NY, October 1999, pp. 475-484.

.... NCP (cf. 3] the question for MDP was open until recently being resolved in the armative by Vardy (cf. 15] Furthermore, the NP hardness of approximating NCP to within any constant factor 8 was proved in [2] whereas MDP was proved NP hard to approximate within any constant only recently (cf. [6]) However, to the best of our knowledge, the exact relationship between the complexity of these two fundamental coding problems, has never been investigated. We prove a result for coding problems analogous to the result on lattices: approximating the Minimum Distance of a code is not harder than ....

....factor and the rank of the lattice, and can be adapted to other problems with similar structure, like the Minimum Distance Problem and the Nearest Codeword Problem for linear codes. In both cases, we reduced a homogeneous problem to the corresponding inhomogeneous one. 10 The results in [12] and [6] are in a certain sense a converse to our result. In [12] and [6] the Shortest Vector Problem and the Minimum Distance Problem are proved NP hard to approximate by reduction from the Closest Vector Problem and the Nearest Codeword Problem respectively. Therefore the inhomogeneous problem is ....

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I. Dumer, D. Micciancio, M. Sudan. Hardness of approximating the minimum distance of a linear code, to appear in Proc. 40th Symposium on Foundations of Computer Science (FOCS '99), IEEE Computer Society, New York, NY, October 1999.

....the code be speci ed If it is part of the input, almost immediately hardness results crop up. The rst such results were due to Berlekamp, McEliece and van Tilborg [13] Subsequently many variants have been shown to remain hard e.g. approximation [2, 20] to within error bounded by distance [23], xed number of errors [21] See also the survey by Barg [5] In this section we will not deal with such problems, but focus on this problem for xed classes of (algebraic) codes. Even after we x the code (family) to be decoded, the decoding problem is not completely xed. The literature ....

Ilya Dumer, Daniele Micciancio, and Madhu Sudan. Hardness of approximating the minimum distance of a linear code. Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pages 475-484, New York City, New York, 17-19 October, 1999.

....not apply for codes over any xed size alphabet, and in particular for binary codes. The quantity L poly ( is even less well understood. When is either very large (of the form 1=2 o(1) or very small (of the form o(1) there is some evidence con rming this bound. In particular, Dumer et al. [6] construct a family of linear codes C, for any 0, for which (n) n 1 and L poly ( 2 ) which matches the conjecture above reasonably closely. We give a simple probabilistic argument to show Theorem 10 For every 0, L poly ( 1 2 (1 n 1=2 ) 1 2 (1 1 3 n 1=2 ) ....

I. Dumer, D. Micciancio and M. Sudan. Hardness of approximating the minimum distance of a linear code. Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), New York, NY, October 1999, pp. 475-484. 21

.... (cf. BMT] the question for MDP was open until recently being resolved in the affirmative by Vardy (cf. V] Furthermore, the NP hardness of approximating NCP to within any constant factor was proved in [ABSS] whereas MDP was proved NP hard to approximate within any constant only recently (cf. [DMS]) However, to the best of out knowledge, the exact relationship between the complexity of these two fundamental coding problems, has never been investigated. We prove a result for coding problems analogous to the result on lattices: approximating the Minimum Distance of a code is not harder than ....

....(approximately) solve the decoding problem (for linear codes) exists, then we can also efficiently find good codes. Interestingly, algebraic geometry codes performing better than the Gilbert Varshamov bound have been used to prove the NP hardness of approximating the Minimum Distance Problem (cf. [DMS]) The reduction from MDP to NCP is basically the same as the lattice one. Actually, it is even easier to establish for binary codes, as the analogue of Proposition 3.1 is trivial (and in fact holds for any non zero codeword) Eq. 1) simplifies too, since here multiplying a column by 2 results in ....

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I. Dumer, D. Micciancio, M. Sudan. "On the hardness of approximating the minimum distance of a linear code", Manuscript.

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I. Dumer, D. Micciancio, and M. Sudan. Hardness of approximating the minimum distance of a linear code. In IEEE Transaction on Information Theory, 49(1), 2003.

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I. Dumer, D. Micciancio and M. Sudan, Hardness of approximating the minimum distance of a linear code, IEEE Transactions on Information Theory, vol. 49(1), 2003, pp. 22--37.

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Ilya Dumer, Daniele Micciancio, and Madhu Sudan. Hardness of approximating the minimum distance of a linear code. In Proceedings of FOCS 1999.

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