A. Barg. Complexity issues in coding theory. In V. S. Pless and W. C. Hu#man, editors, Handbook of Coding theory, volume I, chapter 7, pages 649--754. NorthHolland, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a These parameters a and b will be of major importance in our design. Lemma 11 ( 6] For any s and n, there exists a set Z of at most blog2b decompositions such that any vector x X n has a relatively light subvector xz on some decomposition Z Z . Lemma 11 is discussed in detail in [6] and [4], where the set ( is written out explicitly. Here we omit the lengthy proof. Instead, we consider two examples that illuminate general outline. We also use these examples in the sequel to design two different implementations for a binary (63, 30) code. Let , be a decomposition of the subblock ....

A. Barg, "Complexity issues in coding theory," Handbook of Coding Theory, Ed. V. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998.

....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 includes a multitude of decoding problems: Maximum likelihood ....

Alexander Barg. Complexity issues in coding theory. In [49, Chapter 7].

....are not explicit. We conclude this section by mentioning some papers on peripherally related codes. Codes for correcting asymmetric and unidirectional errors are discussed in [BR82] Et91] EO98] WVB88] and [WVB89] Erasure correcting codes are discussed by Alon and Luby [AL96] and Barg [Ba98] Acknowledgements I would like to thank Andries Brouwer, Suhas Diggavi, Andrew Odlyzko and Eric Rains for conversations about the subject of this paper; and David Applegate, David Johnson and Mauricio Resende for their help in establishing that the V T 0 (n) codes are optimal for n # 9 and ....

A. Barg, Complexity issues in coding theory, pp. 649--754 in Handbook of Coding Theory, ed. V. S. Pless and W. C. Hu#man, North-Holland, Amsterdam, 1998.

.... fairly recently only one class of codes was known to achieve a non zero decoding exponent in polynomial time (though less than E(R; p) for rates arbitrarily close to channel capacity: the class of concatenated codes, introduced by Forney [8] and extensively studied through the mid eighties (see [1], 6] for overviews) In the nineties the discovery of turbo codes [3] with their largely unexplained close tocapacity performance shifted emphasis to iterative decoding techniques. One particular class of codes that can be iteratively decoded is that of expander codes. An expander code is ....

A. Barg, \Complexity issues in coding theory," in Handbook of Coding Theory (V. Pless and W.C. Human, Eds.), Vol.I, Amsterdam: Elsevier Science (1998), pp. 649-754.

.... both nonvanishing rate and distance and low construction complexity [11] 5] Improvements of the parameters of these families are based on multilevel concatenations [3] and the use of algebraic geometry codes [6] A detailed account of the early work is given in [3] see also overviews in [4] [1]. For a given rate R 2 [0; 1] denote by ffi 0 = ffi 0 (R) the GV distance, i.e. the smaller root of the equation R = ln q Gamma H q (ffi) where H q (x) x ln(q Gamma 1) Gamma x ln x Gamma (1 Gamma x) ln(1 Gamma x) is the entropy function. Suppose the constituent codes A and B are both ....

A. Barg, Complexity issues in coding theory, Handbook of Coding Theory (V. Pless and W. C. Huffman, eds.), vol. 1, Elsevier Science, Amsterdam, 1998, pp. 649--754.

.... both nonvanishing rate and distance and low construction complexity [11] 5] Improvements of the parameters of these families are based on multilevel concatenations [3] and the use of algebraic geometry codes [6] A detailed account of the early work is given in [3] see also overviews in [4] [1]. For a given rate R 2 [0; 1] denote by ffi 0 = ffi 0 (R) the GV distance, i.e. the smaller root of the equation R = ln q Gamma H q (ffi) where H q (x) x ln(q Gamma 1) Gamma x ln x Gamma (1 Gamma x) ln(1 Gamma x) is the entropy function. Suppose the constituent codes A and B are both ....

A. Barg, Complexity issues in coding theory, Handbook of Coding Theory (V. Pless and W. C. Huffman, eds.), vol. 1, Elsevier Science, Amsterdam, 1998, pp. 649--754. 7

....in this paper means a ary linear code. Below we frequently refer to random linear codes. By this we mean codes defined by parity check matrices with independent entries chosen from with uniform distribution. Metric properties of this ensemble of codes have been studied relatively well (see, e.g. [4] for a review) We shall study asymptotic properties of this ensemble, meaning that we estimate certain parameters of codes of a given length , then allow to grow, and study the asymptotic behavior of the estimates. Usually as , a property under consideration is valid for all codes in the ensemble ....

....If later it fails to be chosen, this is a part of the inherent error event rather than the fault of a specific decoder. Note that in practice we often do not need to store the whole list, keeping only the most plausible candidate obtained so far. More details and a general overview are found in [4]. This paper is organized as follows. In the next section we overview known results, with which we later compare our algorithm. Since our algorithm uses the ideas of two known methods, covering set decoding [11] 24] and split syndrome decoding [14] we discuss these algorithms in a greater ....

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A. Barg, "Complexity issues in coding theory," in Handbook of Coding Theory, vol. 1, V. Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: Elsevier Science, 1998, pp. 649--754.

....given a nonminimal code vector (z in our case) and a vector y 2 X(D(0) y 2 D(z) we can always cast it away so that the remaining subset of zero neighbours still satisfies condition (13) Therefore Z min can be chosen to be a subset of M. For more details and a general overview we refer to [4]. Remarks. i) Generally, not all zero neighbours are minimal. Indeed, consider the code f0000; 1100; 0011; 1111g. Then vector 0110 lies equally far from all the code vectors which proves that all nonzero code vectors are zero neighbours. However, the all one vector is not minimal. Looking at ....

....min is formed by minimal code vectors. ii) In view of Theorem 4.7, the set Z min is in the general case unavoidable in gradient like decoding methods. For this reason it is no surprise that in the case of arbitrary q the zero17 neighbours algorithm is also applicable and leads to similar results [4]. Interestingly, minimal vectors do not always form a test set in q ary linear codes. 5 Secret sharing A general introduction to secret sharing schemes can be found for instance in Stinson s survey article [21] Some familiarity with this concept is helpful in reading this section. The relation ....

A. Barg, "Complexity issues in coding theory," in V. Pless and W. Cary Huffman, Eds., Handbook of Coding Theory, Elsevier Science, to be published.

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A. Barg. Complexity issues in coding theory. In V. S. Pless and W. C. Hu#man, editors, Handbook of Coding theory, volume I, chapter 7, pages 649--754. NorthHolland, 1998.

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A. Barg, Complexity issues in coding theory, 649-754. Handbook of Coding Theory , R. Brualdi, W. C. Human, and V. Pless, (1998). Elsevier, Amsterdam,

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