N. Alon, J. Bruck, J. Naor, M. Naor, and R.M. Roth. Construction of asymptotically good low-rate errorcorrecting codes through pseudo-random graphs. IEEE Trans. on Information Theory, Vol. 38, No. 2, pp. 509-- 516, March, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....yields an algorithm that is deterministic. Like the randomized algorithm, it is logn attribute efficient. However, it does have a higher query complexity in r and s than the randomized algorithm. Our deterministic construction is based on a class of error correcting codes developed by Alon et al. [2] that have asymptotically good low rate. The construction is similar to a previous construction of Naor et al. 16] for (n; k; r) splitters of small size. Deterministic Construction Given an alphabet Sigma, a code of length m over the alphabet Sigma is a set C Sigma . The elements of C ....

....i . The minimal relative distance of a code C is min u;v2C and u6=v d(u; v) m : We now prove the following lemma. Lemma 4 For any constant 0 1 there is an explicit construction of an (n; r; r ; O(r log r log n) coloring set. Proof: Let k = r= 1 Gamma . We use the code in [2]. This code is n code words v 1 : v n of length L = O(k log k log n) over the alphabet [k ] with a minimal relative distance of at least 1 Gamma 2=k . Now define the coloring A = fa 1 ; aL g [r] where a i;j = v j;i . We now show that this coloring is an (n; r; r ; ....

N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth, Construction of asymptotically good low-rate error-correcting codes through pseudorandom graphs, IEEE Transactions on Information Theory, 38:509-516, 1992.

....to the neighbors of the i th node on the left side of G. Now, the nal codeword (of length n) is obtained by juxtaposing or concatenating the symbols received at each of the n vertices on the right. The construction scheme is similar in spirit to earlier expander based code constructions in [1, 2, 9], and speci cally the construction of near MDS erasure codes in [2] We now elaborate a bit on how we pick each of these components. The left code C will be a linear time code of rate very close to one, say, 1 ) for some small 0, which can correct a fraction ( of errors in linear ....

Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ronny Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38:509-516, 1992.

....implies that a conjunction of O(log n= log log n) number of O(log n) term DNF is learnable in polynomial time in n. Proof. An (n; k) Gammauniversal set is a set fb 1 ; b t g f0; 1g such that every subset of k variables assumes all of its 2 possible assignments in the b i s. In [ABNR92,CZ93] an (n; k) Gammauniversal set is constructed of size t = O(k2 3k log n) This size is polynomial when k = O(log n) We show that the above classes are subsets of (A) where A is any (n; k) universal set. Let f be any k CNF. Let C = X c 1 i 1 Delta Delta Delta X c l i l , l k be a clause ....

Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ron M. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs, IEEE Transactions on information theory, 38, 2:509-516, 1992.

....every satisfying 1. One can use such codes as inner codes in a concatenated scheme with outer code being any code of ) and relative distance 1 O( like Reed Solomon codes, or, if one wants codes over constant alphabet size, algebraic geometric codes or the expander based codes of [2]) Such a concatenation scheme gives a uniformly constructive family of binary linear codes of = lg(1= that can be list decoded from a fraction (1 ) of erasures using lists of size O(1= The previous best result in this vein from [19] achieved a rate of ) by exploiting the fact ....

....of ) by exploiting the fact that any family of codes with relative distance greater than (1 ) 2 has a small list size for decoding from a fraction (1 ) erasures (see Section 2. 1 for some bounds of this nature) The best construction of such high distance codes achieves a rate of ) see [2]) We stress that the novelty of our result is that it is not obtained by appealing to this distance to ELDR relation, and indeed no polynomial time constructions of binary code families of relative distance (1=2 O( and rate about are known. In fact such a construction (which will ....

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N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. on Information

....in a concatenation scheme to give explicit construction of codes with n = O(k=ffi 3 ) but these codes inherit the high construction complexity of algebraic geometric codes. The best code construction of reasonable complexity that has relative distance (1 Gamma 1=q) 1 Gamma ffi) is due to [3], and this also achieves n = O(k=ffi 3 ) For the case of errors, Zyablov and Pinsker [34] see also [7] show that binary linear codes of rate Omega Gamma fl 2 ) exists that have a small number of codewords in any Hamming ball of radius (1=2 Gamma fl) and thus can, in principle, be list ....

....can be recovered in polynomial time when up to a fraction (1 Gamma fl) of the symbols in the received word are erased. 4 Proof: By Corollary 1, we only need to construct a code with dimension k, blocklength n and minimum distance (1 Gamma 1 q ) 1 Gamma fl)n. As shown by Alon et al. [3], such a code can be constructed in polynomial in n time (with the exponent being independent of fl) provided n = Omega Gamma k=fl 3 ) We could have similarly used the code C AG GammaHad (k; fl) from Proposition 3. 2 4 Both the construction of the code and the list decoding can be performed ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. on Information Theory, 38 (1992), pp. 509-516.

....such a large relative distance achieve a rate of ) and 3 ) respectively. However, such codes either use algebraic geometric codes [9] and hence su er from complicated decoding procedures (in particular the decoding time is at least cubic in the blocklength) or as in the constructions of [1], are explicitly constructible but lack a polynomial time algorithm to decode up to half the minimum distance (or for that matter to decode up to any positive constant, no matter how small, fraction of errors) University of California at Berkeley, Computer Science Division, Berkeley, CA ....

....transform a heavily corrupted codeword y of C 1 to a much less corrupted codeword x of C. The latter can be decoded using the linear time decoding algorithm for C. The encoding can also be accomplished in linear time using the linear time encoder for C. Our constructions are similar to those in [1] where expanders were used to amplify the distance and thus construct codes of large minimum distance. However, while the results of [1] use only the vertex expansion properties of the graph G, here we use much stronger isoperimetric properties o ered by expander graphs. We need these stronger ....

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N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. on Information Theory, 38 (1992), pp. 509-516.

....(AG) codes, but with much faster encoding and decoding algorithms. Codes of rate ) over constant sized alphabet that can be uniquely decoded from (1=2 ) errors in near linear time (once again this matches AG codes with much faster algorithms) This construction is similar to that of [1], and our decoding algorithm can be viewed as a positive resolution of their main open question. Linear time encodable and decodable binary codes of positive rate 1 (in fact, rate 4 ) that can correct up to (1=4 ) fraction errors. Note that this is the best error correction one can ....

....decoding and construction algorithms. In comparison, our decoding algorithms are simple and have running times arbitrarily close to linear; however, the constant in the rate is affected by the exponent in the running time. We also mention that our codes are very similar to the codes constructed in [1]. In the latter paper the authors asked if there is a polynomial time decoding algorithm for their codes; thus our results can be viewed as positive resolution of the main open problem from their paper. Our last class of codes comes with linear time encoding and decoding algorithms (call such ....

[Article contains additional citation context not shown here]

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. on Information Theory, 38 (1992), pp. 509-516.

.... satisfying 1. One can use such codes as inner codes in a concatenated scheme with outer code being any code of rate ( and relative distance 1 O( like Reed Solomon codes, or, if one wants codes over constant alphabet size, algebraic geometric codes or the expander based codes of [2]) Such a concatenation scheme gives a uniformly constructive family of binary linear codes of rate ( 2 = lg(1= that can be list decoded from a fraction (1 ) of erasures using lists of size O(1= 1 The previous best result in this vein from [14] achieved a rate of ( 3 ) by ....

....result in this vein from [14] achieved a rate of ( 3 ) by exploiting the fact that any family of codes with relative distance greater than (1 ) 2 has a small list size for decoding from (1 ) erasures. 2 The best construction of such high distance codes achieves a rate of ( 3 ) see [2]) We stress that the novelty of our result is that it is not obtained by appealing to this distance to ELDR relation, and indeed no polynomial time constructions of binary code families of relative distance (1=2 O( and rate about 2 are known. In fact such a construction (which will ....

[Article contains additional citation context not shown here]

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. on Information Theory, 38 (1992), pp. 509-516.

.... codes than those obtained in [7] The codes constructed in [34] are concatenated codes (see below) 4 ALICE SILVERBERG, JESSICA STADDON, AND JUDY WALKER where the outer code is an algebraic geometry code coming from a Hermitian curve, while those used in [7] come from pseudo random graphs (see [1]) 2. Background on Codes and Traceability In this section we give definitions, notation, and background on codes, traceability, and the decoding techniques that form the basis for our tracing algorithms. 2.1. Definitions and Notation. A code C of length r is a subset of Q r , where Q is a ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory 38 (1992), 509--516.

....codes as outer codes in a concatenation scheme to give explicit construction of codes with n = O(k= 3 ) but these codes inherit the high construction complexity of algebraic geometric codes. The best code construction of reasonable complexity that has relative distance (1 1=q) 1 ) is due to [3], and this also achieves n = O(k= 3 ) For the case of errors, Zyablov and Pinsker [34] see also [7] show that binary linear codes of rate 2 ) exists that have a small number of codewords in any Hamming ball of radius (1=2 ) and thus can, in principle, be list decoded up to this radius. ....

....list of possible codewords can be recovered in polynomial time when up to a fraction (1 ) of the symbols in the received word are erased. 4 Proof: By Corollary 1, we only need to construct a code with dimension k, blocklength n and minimum distance (1 1 q ) 1 )n. As shown by Alon et al. [3], such a code can be constructed in polynomial in n time (with the exponent being independent of ) provided n = k= 3 ) We could have similarly used the code C AG Had (k; from Proposition 3. 2 4 Both the construction of the code and the list decoding can be performed in time which ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. on Information Theory, 38 (1992), pp. 509-516.

....2c for L = 8c log n. Combining this with the code 0 0 (2c) and Lemma 3.2 we get a c frameproof code for n users whose length is 2cL = 16c 2 log n. 3 To make this construction explicit we must use an explicit low rate error correcting code. Explicit constructions of such codes are described in [1]. The explicit construction are not as good as the bounds provided by Lemma 3.3. Using a simple explicit low rate code it is possible to obtain codes of length l = c 2 log 2 (n) 4 c secure Codes We now turn our attention to the full problem of collusion secure fingerprinting. Suppose a ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of asymptotically good low-rate errorcorrecting codes through pseudo-random graphs, IEEE Transactions on Information Theory, vol. 38, 1992, pp. 509--516.

....of an (n; n] k ) universal set of size O(2 k k log n) for fixed k, which is optimal following the lower bound argument in [SB88] Alon [Alo86] showed an explicit construction of (n; n] k ) universal sets of size log n Delta 2 O(k 2 ) for the case k is fixed. In [NN90] and [ABNNR92], nearly optimal constructions of size log n Delta 2 O(k) were given. The previously best known construction in this case is due to [NSS95] and has size 2 k k O(logk) log n. Our construction has the least size and at the same time matches the bound given by the probabilistic argument. 100 ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. on Info. Theory, 38(2):509-515, 1992.

....s. Let C be a code C 0, 1 GF[Q] with the properties: Q is roughly equal to [ n is roughly equal to nQ a rn,rn2, with rn rn2, C(rn) and C(rn2) differ in at least 1 p fraction of their entries. The best known construction for such a code C is described by Alon et al. in [1]. The shared string s = i,a,b) consists of three random values where: i r 1. n , a r GF[Q] O , br GF[Q] Using those three values s(rn) is evaluated as: s(rn) aCt(m) b. The single round protocol Pt is: 36O P: A Sound Single Round, Secrecy Flog(n) 511og(p x ) 1] Probability p ....

N. Alon, J. Bruck, J. Naor, M. Naor, R. Roth, Construction of Asymptotically Good Low-Rate Error-Correcting Codes through Pseudo. Random Graphs, IEEE Transactions on Information Theory, Vol. 38, No. 2, March 1992

....length , over an alphabet of size 2k 2 , such that the distance between every two codewords is at least =k 2 . The goal is to construct such a code with as small as possible. There are no known explicit constructions that match the probabilistic bound. For the best known construction see [1] and references therein. For small k the constructions of [1] yield a scheme with m = O(k 6 log n) and r = O(k 8 log n) 4.2 An Open Two Level Scheme The two level traceability scheme, described in this subsection, can be thought of as iterating the previous construction two times. While ....

....between every two codewords is at least =k 2 . The goal is to construct such a code with as small as possible. There are no known explicit constructions that match the probabilistic bound. For the best known construction see [1] and references therein. For small k the constructions of [1] yield a scheme with m = O(k 6 log n) and r = O(k 8 log n) 4.2 An Open Two Level Scheme The two level traceability scheme, described in this subsection, can be thought of as iterating the previous construction two times. While it is more complicated than the simple scheme, it saves about ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of Asymptotically Good Low-Rate Error-Correcting Codes through Pseudo-Random Graphs, IEEE Transactions on Information Theory, vol. 38 (1992), 509-516.

....in the first method (Theorem 3.1) whereas all known methods for using (traditional linear codes) are more complicated and require exponentiation. Using our techniques we can actually enhance the error correction of a linear code without decreasing the rate by much. This is further investigated in [6]. 3.2 Sampling the family F The problem of obtaining the vectors r 1 ; r l in Stage 2 with the desired property can be abstracted in the following way. Suppose that there is a universe, and we wish to find a member in a certain subset S of it. In our case it is the set of vectors for ....

N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, Construction of Asymptotically Good Low-Rate Error-Correcting Codes through Pseudo-Random Graphs, IEEE Transactions on Information Theory, vol. 38 (1992), pp. 509-516.

....to decode the message from the encoding is proportional only to the amount of the message that is missing. Our scheme is based on the properties of expanders which are explicit 3 graphs with pseudo random properties. The relevance of these graphs to error correcting codes has been observed in [5], and indeed we apply some of the ideas of that paper. Erasure resilient codes are related to error correcting codes, and are typically easier to design. For example, an error correcting code with encoding length cn that can correct up to bn bit flips can be used as an erasure resilient code that ....

N. Alon, J. Bruck, J. Naor, M. Naor, R. Roth, "Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs", IEEE Transactions on Information Theory, Vol. 38, 1992, pp. 509-516.

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N. Alon, J. Bruck, J. Naor, M. Naor, and R.M. Roth. Construction of asymptotically good low-rate errorcorrecting codes through pseudo-random graphs. IEEE Trans. on Information Theory, Vol. 38, No. 2, pp. 509-- 516, March, 1992.

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Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ronny Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38:509-516, 1992.

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N. Alon, J. Bruck, J. Naor, M. Naor, and R.M. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38(2):509-516, 1992.

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Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ronny Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38:509-516, 1992.

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N. Alon, J. Bruck, J. Naor, M. Naor and R. Roth, "Construction of Asymptotically Good, Low-Rate Error-Correcting Codes through Pseudo-Random Graphs", IEEE Transactions on Information Theory 38 (1992), pp. 509--516. 23

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N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. Info. Thy., 38(2):509--512, March 1992.

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Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ronny Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38:509--516, 1992.

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Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ronny Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38:509-516, 1992.

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N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. Info. Thy., 38(2):509--512, March 1992.

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