A. Vardy, "The intractability of computing the minimum distance of a code," IEEE Trans. Inform. Theory, vol. 43, pp. 1757--1766, November 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....subject was introduced in some classical papers [4] 8] 10] Many other papers deal with the subject, covering methods to compute the complexity parameters, build the trellis, and establish bounds, or studies on the complexity reduction by coordinate permutation. Among them we can cite [11] [26]. Recent results on the complexity of convolutional codes are reported in [17] and [27] For a complete list of references on the subject, see [28] B. Minimal Span Generator Matrix (MSGM) Matrices A procedure for practically computing the state and branch profile of a code has been presented in ....

....Example 2: Given the code of Example 1, the following permutation: leads to . The generator matrix has the LR property. It follows We have . The minimal trellis of is depicted in Fig. 1(b) In general, the problem of finding the best permutation for complexity minimization is NP complete [25] [26]. D. Sectionalization So far, we have considered an atomic representation of , i.e. a bit at every time . For clearness, in the following we will denote by , and , the complexity parameters obtained when a sectionalization occurs such that a group of bits corresponds to any time . The ....

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A. Vardy, "The intractability of computing the minimum distance of a code," IEEE Trans. Inform. Theory, vol. 43, pp. 1757--1766, Nov. 1997.

....with cryptanalytical tools and knowledge which can e ectively nd the best approximation in any sense over any given S box. Unfortunately, this is not the case in general since Vardy has recently proved the long time hypothesis that determining the minimum distance of a linear code is NP complete [25]. However, in the following section we show that it is sometimes possible to nd good approximations over S boxes using algorithms for decoding. 96 2 14 12 11 4 2 1 12 7 4 10 7 11 13 6 1 8 5 5 0 3 15 15 10 13 3 0 9 14 8 9 6 4 11 2 8 1 12 11 7 10 1 13 14 7 2 8 13 15 6 9 15 12 0 5 9 6 10 3 4 0 ....

Alexander Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. on Info. Th., vol. 43, no. 6, Nov. 1997.

....distance. No such algorithm is known. The complexity of this problem (can it be solved in polynomial time or not ) was rst explicitly questioned by Berlekamp, McEliece and van Tilborg [7] in 1978 who conjectured it to be NP complete. This conjecture was nally resolved in the armative by Vardy ([17]) in 1997. 17] also gives further motivations and detailed account of prior work on this problem. To advance this search of good codes, one can allow for approximate solution, by nding code distance within some interval. In this paper, we study this problem and show that it is hard to ....

....algorithm is known. The complexity of this problem (can it be solved in polynomial time or not ) was rst explicitly questioned by Berlekamp, McEliece and van Tilborg [7] in 1978 who conjectured it to be NP complete. This conjecture was nally resolved in the armative by Vardy ( 17] in 1997. [17] also gives further motivations and detailed account of prior work on this problem. To advance this search of good codes, one can allow for approximate solution, by nding code distance within some interval. In this paper, we study this problem and show that it is hard to approximate the minimum ....

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A. Vardy. \The Intractability of Computing the Minimum Distance of a Code," IEEE Trans. Inform. Theory, Vol. IT-43, no. 6, November 1997, pp. 1757-1766.

....No such algorithm is known. The complexity of this problem (can it be solved in polynomial time or not ) was first explicitly questioned by Berlekamp, McEliece and van Tilborg [7] in 1978 who conjectured it to be NP complete. This conjecture was finally resolved in the affirmative by Vardy ([15]) in 1997. 15] also gives further motivations and detailed account of prior work on this problem. We examine the approximability of this parameter and show that it is hard to approximate the minimum distance to within any constant factor, unless NP = RP (i.e. every problem in NP has a ....

....is known. The complexity of this problem (can it be solved in polynomial time or not ) was first explicitly questioned by Berlekamp, McEliece and van Tilborg [7] in 1978 who conjectured it to be NP complete. This conjecture was finally resolved in the affirmative by Vardy ( 15] in 1997. [15] also gives further motivations and detailed account of prior work on this problem. We examine the approximability of this parameter and show that it is hard to approximate the minimum distance to within any constant factor, unless NP = RP (i.e. every problem in NP has a polynomial time ....

[Article contains additional citation context not shown here]

A. Vardy. "The Intractability of Computing the Minimum Distance of a Code," IEEE Trans. Inform. Theory, Vol. IT-43, no. 6, November 1997, pp. 1757--1766.

....[11, Proposition 6.5] that, for a connected matroid, if t i;j 0, then t k;l 0 for all (k; l) 6= 0; 0) such that 0 k i and 0 l j. Hence = maxfj : t 1;j 0g: Thus we have reduced the problem to nding the minimum distance of a linear code over F q and, by the recent result of Vardy [36], we know this is NP hard for all prime powers q. Note 9.10 (i) We cannot replace an F q representable matroid by a graphic matroid (or even a regular matroid) in the above statement. ii) By Lemma 1.1, the last theorem implies that, unless NP = RP , there is no fpras for t 1;w for general ....

A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory 43 (1997), 1757-1766.

....see e.g. ABSS] Van Emde Boas has proved the NP hardness of the nearest vector problem in all L p norms 1 p 1. Arora, Babai, Stern and Sweedyk (cf. ABSS] has shown that even to approximate the nearest vector within a constant factor is NP hard. A. Vardy has proved recently (cf. [V1] or [V2] that deciding whether there is a codeword within a given distance is NP complete. The interested reader may find more detailed information about these and other related problems in [ABSS] or [V2] 5. The motivation for proving that to find short vectors in lattices is hard, in some ....

A. Vardy, The Intractability of Computing the Minimum Distance Code, Preprint

....n. If for a constant c 0, it holds that m 2 n=c , then Worst case DLC can be solved in polynomial time. Going through all possible 2 n solutions requires time at most m c . Chapter 1. Introduction 17 A survey of positive and negative results on the hardness of decoding can be found in [Var97, Section 2.2] in particular for families of linear codes, that is, for linear codes with additional properties that are useful in practice. From this survey, it stands out that worst case polynomial time maximum likelihood decoding algorithms are not known for any specific family of useful ....

....m , unless NP QP, the class of deterministic quasi polynomial time algorithms. A problem related to the maximum likelihood decoding problem is the minimum distance of a code problem (as defined at the end of Section 1.3. 1, Definition 14) The corresponding decision problem is NP complete [Var97] and the functional problem is hard to approximate in the two following ways [DMS99] Theorem 25 [Hardness of approximation of the minimum distance of a code] Let ffi 0. For an n Theta m code A, let the minimum distance be d . Then there is no random polynomial time algorithm which ....

A. Vardy. The intractability of computing the minimum distance of a code. IEEE Trans. Information Theory, IT-43(6), November 1997.

.... weight, asking for a vector of weight at least w is also NPcomplete for binary codes [6] Finally, the problem Codeword of minimal weight for binary linear codes, asking for a non zero vector of weight at most w was conjectured NP complete in [1] and finally proven to be so in a recent paper [9]. These problems are equivalent to asking if the linear space generated by the columns of H contain a non zero vector of weight w, at least w, or at most w. They are thus very close to (EVEN, EVEN) set problems. However, inputs to the (EVEN, EVEN) set problem are square matrices, and for ....

A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory. Vol.43 No. 6, 1997, 1757-1766. Preliminary version STOC'97.

....No such algorithm is known. The complexity of this problem (can it be solved in polynomial time or not ) was first explicitly questioned by Berlekamp, McEliece and van Tilborg [7] in 1978 who conjectured it to be NP complete. This conjecture was finally resolved in the affirmative by Vardy ([16]) in 1997. 16] also gives further motivations and detailed account of prior work on this problem. We examine the approximability of this parameter and show that it is hard to approximate the minimum distance to within any constant factor, unless NP = RP (i.e. every problem in NP has a ....

....is known. The complexity of this problem (can it be solved in polynomial time or not ) was first explicitly questioned by Berlekamp, McEliece and van Tilborg [7] in 1978 who conjectured it to be NP complete. This conjecture was finally resolved in the affirmative by Vardy ( 16] in 1997. [16] also gives further motivations and detailed account of prior work on this problem. We examine the approximability of this parameter and show that it is hard to approximate the minimum distance to within any constant factor, unless NP = RP (i.e. every problem in NP has a polynomial time ....

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A. Vardy. "The Intractability of Computing the Minimum Distance of a Code," IEEE Trans. Inform. Theory, Vol. IT-43, no. 6, November 1997, pp. 1757--1766.

....provided the field over which the code is specified is not fixed; however, we are able to fix the characteristic of the field. We leave open the problem of dealing with the case of a fixed field, in particular GF (2) Our proof uses several ideas from the recent breakthrough result of Vardy [12] showing the NPhardness of the problem of computing the minimum distance of a binary linear code thus settling a conjecture of Berlekamp, McEliece and van Tilborg dating back to 1978. In particular, we use his ingenious construction for obtaining MDS codes using Vandermonde matrices. Vardy also ....

....far, there is no general agreement on what is the best measure of trellis complexity (see [8] for a recent position in this matter) As we shall see, the intractability result of this paper holds for any choice of measure from the above list. 3 NP hardness of Restricted MDS Code Recently, Vardy [12] proved that it is NP hard to find whether or not a given linear code is MDS. We show that the problem remains NP hard even when restricted to (2k; k) linear codes; this restricted version of the MDS Code problem will be used to derive the results of the next section. Problem: Restricted MDS Code ....

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A. VARDY, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory, vol.43, pp. 1757-1766, 1997.

....on NP complete problems related to coding was Berlekamp et al. 28] The results of this paper correspond to parts (a) and (c) of our Theorem 4.1. That Weight of error is NP complete for arbitrary q and for multilevel codes was observed in Barg [19] Part (b) is a recent breakthrough due to Vardy [161]. The NP completeness of the Minimal weight problem has been conjectured in Berlekamp et al. 28] and has resisted all attacks for 19 years despite repeated calls for a proof; see Johnson [88] Some partial results were proved in Ntafos and Hakimi [124] The NP completeness of deciding whether a ....

....and Shor (see [47] The same result for ternary codes and weight n is from Barg [19] For the NP completeness of P 5 ; P 6 ; P 7 see Frances and Litman [65] Horn and Kschischang [84] and Kratochvil [100] respectively. An easy reduction for P 6 follows from part (b) of Theorem 4. 1; see Vardy [161]. The construction of codes from graphs was introduced in Calabi [34] Hakimi [81] Properties of classical linear spaces in graphs and their combinatorial evolution form the subject of matroid theory; see Welsh [165] Many problems that are difficult for general linear codes, are polynomial for ....

A. Vardy, "The intractability of computing the minimum distance of a code," IEEE Trans. Inform. Theory, to appear. Preliminary version in Proc. 29th ACM Annual Sympos. on the Theory of Computing (STOC'97), ACM (1997), pp. 92-- 109.

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A. Vardy, "The intractability of computing the minimum distance of a code," IEEE Trans. Inform. Theory, vol. 43, pp. 1757--1766, November 1997.

....Distance requires at most k columns in a solution. Berlekamp, McEliece and van Tilborg [BMvT78] proved that Maximum Likelihood Decoding and Weight Distribution are NP complete, by means of a reduction from 3 Dimensional Matching. They conjectured that MinimumDistance is also NP complete, and Vardy[Var97b] recently proved this conjecture using a non parametric reduction from Maximum Likelihood Decoding. Since 3 Dimensional Matching is fixed parameter tractable, these earlier results do not allow us to conclude anything about the parameterized complexity of the three problems. Over the past few ....

.... shown that four of six fundamental computational problems in the domains of linear codes and integer lattices are NP complete and hard for the parametrized complexity class W [1] The obvious outstanding open problems are: ffl Is the Minimum Distance problem, recently proved to be NP complete in [Var97b], also hard for W [1] ffl Is the Shortest Vector problem hard for NP and W [1] A consequence of the proof in [Var97b] that the Minimum Distance problem is NP hard is that Even Set is NP hard. This leaves us in the curious situation that the only known proof of this seemingly quite combinatorial ....

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A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory, 43, (1997), 1757--1766. 28

....algorithm of [55] with respect to the same optimality criterion, then the resulting decision diagrams will be isomorphic. The converse is also true: if two sectionalized decision diagrams are isomorphic, they represent the same function. Complexity of the variable ordering problem. It is known [10, 11, 43, 45, 74] that the variable ordering problem for binary decision diagrams and the permutation problem for trellises are both computationally hard. However, the known NP hardness results establish the intractability of different aspects of these equivalent problems. The primary intractability result in the ....

....length n, specified by its parity check or generator matrix, a positive integer i n, and a positive size bound s. Question: Is there a permutation of the time axis, such that the number of vertices at time i in the corresponding minimal trellis for C is less than s It is furthermore shown in [74] that this problem remains NP complete if the size bound is restricted to s = 2 i . When translated into the context of binary decision diagrams, using Theorem 1, this implies the following result. Suppose we are given a positive integer i n and a Boolean function f(x 1 ; x n ) ....

A. Vardy, "The intractability of computing the minimum distance of a code," IEEE Trans. Inform. Theory, vol. 43, pp. 1757--1766, November 1997.

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A. Vardy. The intractability of computing the minimum distance of a code. In IEEE Transaction on Information Theory, 43(6), 1997.

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A. Vardy. The Intractability of Computing the Minimum Distance of a Code. IEEE Transactions on Information Theory, 43(6):1757--1766, November 1997.

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A. Vardy. The intractability of computing the minimum distance of a code. IEEE Transactions on Information Theory, 43(6):1757--1766, 1997.

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Alexander Vardy. The Intractability of Computing the Minimum Distance of a Code. IEEE Transactions on Information Theory, 43:1757--1766, November 1997.

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