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73
Priority Encoding Transmission
 IEEE Transactions on Information Theory
, 1994
"... We introduce a new method, called Priority Encoding Transmission, for sending messages over lossy packetbased networks. When a message is to be transmitted, the user specifies a priority value for each part of the message. Based on the priorities, the system encodes the message into packets for tra ..."
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Cited by 263 (11 self)
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We introduce a new method, called Priority Encoding Transmission, for sending messages over lossy packetbased networks. When a message is to be transmitted, the user specifies a priority value for each part of the message. Based on the priorities, the system encodes the message into packets for transmission and sends them to (possibly multiple) receivers. The priority value of each part of the message determines the fraction of encoding packets sufficient to recover that part. Thus, even if some of the encoding packets are lost enroute, each receiver is still able to recover the parts of the message for which a sufficient fraction of the encoding packets are received. International Computer Science Institute, Berkeley, California. Research supported in part by National Science Foundation operating grant NCR941610 y Computer Science Department, Swiss Federal Institute of Technology, Zurich, Switzerland. Research done while a postdoc at the International Computer Science Institute...
Hardness of Approximating the Minimum Distance of a Linear Code
, 2003
"... We show that the minimum distance d of a linear code is not approximable to within anyconstant factor in random polynomial time (RP), unless NP (nondeterministic polynomial time) equals RP. We also show that the minimum distance is not approximable to within an additiveerror that is linear in the b ..."
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Cited by 52 (7 self)
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We show that the minimum distance d of a linear code is not approximable to within anyconstant factor in random polynomial time (RP), unless NP (nondeterministic polynomial time) equals RP. We also show that the minimum distance is not approximable to within an additiveerror that is linear in the block length n of the code. Under the stronger assumption that NPis not contained in RQP (random quasipolynomial time), we show that the minimum distance is not approximable to within the factor 2log 1ffl(n), for any ffl> 0. Our results hold for codes over any finite field, including binary codes. In the process we show that it is hard to findapproximately nearest codewords even if the number of errors exceeds the unique decoding radius d/2 by only an arbitrarily small fraction ffld. We also prove the hardness of the nearestcodeword problem for asymptotically good codes, provided the number of errors exceeds (2
Averaging bounds for lattices and linear codes
 IEEE Trans. Information Theory
, 1997
"... Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofa ..."
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Cited by 49 (1 self)
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Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofasimple lemma for linear codes over GF (p) used with plevel amplitude modulation. The relation between the combinatorial packing of solid bodies and the informationtheoretic “soft packing ” with arbitrarily small, but positive, overlap is illuminated. The “softpacking” results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda–Poltyrev result that spherically shaped lattice codes and adecoder that is unaware of the shaping can achieve the rate 1=2 log2 (P=N).
Covering arrays and intersecting codes
 Journal of Combinatorial Designs
, 1993
"... A tcovering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once. Such arrays are used for example in circuit testing, and one wishes to minimize k for given values of n and t. The case t = 2 was solved by Rkn ..."
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Cited by 25 (0 self)
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A tcovering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once. Such arrays are used for example in circuit testing, and one wishes to minimize k for given values of n and t. The case t = 2 was solved by Rknyi, Katona, and Kleitman and Spencer. The present article is concerned with the case t = 3, where important (but unpublished) contributions were made by Busschbach and Roux in the 1980s. One of the principal constructions makes use of intersecting codes (linear codes with the property that any two nonzero codewords meet). This article studies the properties of 3covering arrays and intersecting codes, and gives a table of the best 3covering arrays presently known. For large n the minimal k satisfies 3.21256 < k / log n < 7.56444. 01993
On the decoding of algebraicgeometric codes
 IEEE TRANS. INFORM. THEORY
, 1995
"... This paper provides a survey of the existing literature on the decoding of algebraicgeometric codes. Definitions, theorems and cross references will be given. We show what has been done, discuss what still has to be done and pose some open problems. The following subjects are examined in a more or ..."
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Cited by 20 (2 self)
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This paper provides a survey of the existing literature on the decoding of algebraicgeometric codes. Definitions, theorems and cross references will be given. We show what has been done, discuss what still has to be done and pose some open problems. The following subjects are examined in a more or less historical order.
Euclidean Weights Of Codes From Elliptic Curves Over Rings
 TRANS. AMER. MATH. SOC
"... We construct certain errorcorrecting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools; notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. These results may be of independent interest. ..."
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Cited by 13 (5 self)
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We construct certain errorcorrecting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools; notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. These results may be of independent interest.
Decoding AlgebraicGeometric Codes Beyond the ErrorCorrection Bound
, 1998
"... Generalizing the highnoise decoding methods of [1, 19] to the class of algebraicgeometric codes, we design the first polynomialtime algorithms to decode algebraicgeometric codes significantly beyond the conventional errorcorrection bound. Applying our results to codes obtained from curves with m ..."
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Cited by 13 (4 self)
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Generalizing the highnoise decoding methods of [1, 19] to the class of algebraicgeometric codes, we design the first polynomialtime algorithms to decode algebraicgeometric codes significantly beyond the conventional errorcorrection bound. Applying our results to codes obtained from curves with many rational points, we construct arbitrarily long, constantrate linear codes over a fixed field F q such that a codeword is efficiently, nonuniquely reconstructible after a majority of its letters have been arbitrarily corrupted. We also construct codes such that a codeword is uniquely and efficiently reconstructible after a majority of its letters have been corrupted by noise which is random in a specified sense. We summarize our results in terms of bounds on asymptotic parameters, giving a new characterization of decoding beyond the errorcorrection bound. 1 Introduction Errorcorrecting codes, originally designed to accommodate reliable transmission of information through unreliable ...
Sextic double solids
, 2004
"... Abstract. We prove nonrationality and birational superrigidity of a Qfactorial double cover X of P 3 ramified along a sextic surface with at most simple double points. We also show that the condition #Sing(X)  ≤ 14 implies Qfactoriality of X. In particular, every double cover of P 3 with at m ..."
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Cited by 12 (5 self)
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Abstract. We prove nonrationality and birational superrigidity of a Qfactorial double cover X of P 3 ramified along a sextic surface with at most simple double points. We also show that the condition #Sing(X)  ≤ 14 implies Qfactoriality of X. In particular, every double cover of P 3 with at most 14 simple double points is nonrational and not birationally isomorphic to a conic bundle. All the birational transformations of X into elliptic fibrations and into Fano 3folds with canonical singularities are classified. We consider some relevant problems over fields of finite characteristic. When X is defined over a number field F we prove that the set of rational points on the 3fold X is potentially dense if Sing(X) ̸ = ∅.
Decoding geometric Goppa codes using an extra place
 IEEE Trans. Inform. Theory
, 1992
"... Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the code length is smaller than the number of rational points on the curve, then this method can correct up to 1 2 (d ∗ − 1) − s errors, where d ∗ is the designed minimum distance o ..."
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Cited by 12 (3 self)
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Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the code length is smaller than the number of rational points on the curve, then this method can correct up to 1 2 (d ∗ − 1) − s errors, where d ∗ is the designed minimum distance of the code and s is the Clifford defect. The affine ring with respect to a place P is the set of all rational functions which have no poles except at P, and it is somehow similar to a polynomial ring. For a special kind of geometric Goppa codes, namely CΩ(D, mP), the decoding algorithm is reduced to solving the key equation in the affine ring, which can be carried out by the subresultant sequence in the affine ring with complexity O(n3), where n is the length of codewords.
Efficient Traitor Tracing Algorithms using List Decoding
 In Proceedings of ASIACRYPT ’01, volume 2248 of LNCS
, 2001
"... Abstract. We use powerful new techniques for list decoding errorcorrecting codes to efficiently trace traitors. Although much work has focused on constructing traceability schemes, the complexity of the tracing algorithm has received little attention. Because the TA tracing algorithm has a runtime o ..."
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Cited by 12 (0 self)
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Abstract. We use powerful new techniques for list decoding errorcorrecting codes to efficiently trace traitors. Although much work has focused on constructing traceability schemes, the complexity of the tracing algorithm has received little attention. Because the TA tracing algorithm has a runtime of O(N) in general, where N is the number of users, it is inefficient for large populations. We produce schemes for which the TA algorithm is very fast. The IPP tracing algorithm, though less efficient, can list all coalitions capable of constructing a given pirate. We give evidence that when using an algebraic structure, the ability to trace with the IPP algorithm implies the ability to trace with the TA algorithm. We also construct schemes with an algorithm that finds all possible traitor coalitions faster than the IPP algorithm. Finally, we suggest uses for other decoding techniques in the presence of additional information about traitor behavior. 1