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136
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 316 (12 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...
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 97 (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).
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 58 (6 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
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 44 (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
Algebraic Geometric Secret Sharing Schemes and Secure MultiParty Computations over Small Fields
"... We introduce algebraic geometric techniques in secret sharing and in secure multiparty computation (MPC) in particular. The main result is a linear secret sharing scheme (LSSS) de ned over a nite eld Fq, with the following properties. 1. It is ideal. The number of players n can be as large as #C(Fq ..."
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Cited by 43 (9 self)
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We introduce algebraic geometric techniques in secret sharing and in secure multiparty computation (MPC) in particular. The main result is a linear secret sharing scheme (LSSS) de ned over a nite eld Fq, with the following properties. 1. It is ideal. The number of players n can be as large as #C(Fq), where C is an algebraic curve C of genus g de ned over Fq. 2. It is quasithreshold: it is trejecting and t+1+2gaccepting, but not necessarily t + 1accepting. It is thus in particular a ramp scheme. High information rate can be achieved. 3. It has strong multiplication with respect to the tthreshold adversary structure, if t < 1 3 n 4 3 g. This is a multilinear algebraic property on an LSSS facilitating zeroerror multiparty multiplication, unconditionally
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 31 (6 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.
Consecutive Weierstrass gaps and minimum distance of Goppa codes
 J. Pure Appl. Algebra
, 1993
"... Abstract: We prove that if there are consecutive gaps at a rational point on a smooth curve defined over a finite field, then one can improve the usual lower bound on the minimum distance of certain algebraicgeometric codes defined using a multiple of the point. A qary linear code of length n and ..."
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Cited by 23 (0 self)
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Abstract: We prove that if there are consecutive gaps at a rational point on a smooth curve defined over a finite field, then one can improve the usual lower bound on the minimum distance of certain algebraicgeometric codes defined using a multiple of the point. A qary linear code of length n and dimension k is a vector subspace of dimension k of Fnq, where Fq denotes the finite field with q elements. The minimum distance of a code is the minimum number of places in which two distinct codewords differ. The greater the minimum distance, the greater the number of errors that the code can detect or correct. For a linear code, the minimum distance is also the minimum weight of a nonzero codeword, where the weight of a codeword is the number of nonzero places in that codeword. A linear code of length n, dimension k and minimum distance d is called an [n, k, d]code. V.D. Goppa [3,4] realized that one could use the RiemannRoch Theorem to show that certain codes produced from two divisors G and D on a curve have good properties. In particular, he gave lower bounds for the minimum distances of these codes. In a previous article [1], the first and third authors showed that if G is taken to be a multiple of a point P, then knowledge of the gaps at P may allow one to say that the minimum distance of
Toric codes over finite fields
 Appl. Algebra Engrg. Comm. Comput
"... In this note, a class of errorcorrecting codes is associated to a toric variety associated to a fan defined over a finite field Fq, analogous to the class of Goppa codes associated to a curve. For such a “toric code ” satisfying certain additional conditions, we present an efficient decoding algori ..."
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Cited by 19 (1 self)
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In this note, a class of errorcorrecting codes is associated to a toric variety associated to a fan defined over a finite field Fq, analogous to the class of Goppa codes associated to a curve. For such a “toric code ” satisfying certain additional conditions, we present an efficient decoding algorithm for the dual of a Goppa code. Many examples are given. For small q, many of these codes have parameters beating the GilbertVarshamov bound. In fact, using toric codes, we construct a (n,k,d) = (49,11,28) code over F8, which is better than any other known code listed in Brouwer’s tables [B] for that n and k. We conjecture that, for a large subclass of these codes, the parameters (n,k,d) all can be computed by means of an explicitly given formula (depending on q and the toric surface). In general, we conjecture
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 15 (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.