M. A. Tsfafman and S. G. Vladut, Algebraic-Geometric Codes. Dordrecht/Boston/London: Kluwer, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....D P 1 P 2 . P n ,G =uP 0 . Let u 1 0, u 2 , u l , be the pole orders of P 0 in ascending order. Consider the code C = C(D, u l P 0 ) of functions which are everywhere holomorphic except for a pole of degree u l at P 0 , evaluated at P 1 , P n (this is the L construction of [17]) Assume u l n. Then C l has dimension l and the following hold: C has minimum distance u l . Hence u l n. We need curves with many rational points and at least one rational Weierstra point whose gaps are as large as possible. In [14] and [13] a class K q of function ....

M. A. Tsfasman and S. G. Vladut, Algebraic-Geometric Codes, Kluwer, Dordrecht, Boston, London (1991).

....vector y 2 E t (C) By Prop. 3 such a sequence can be obtained from any family of error correcting codes with large minimum distance. A natural candidate is q ary [N; RN ] linear algebraic geometric codes whose relative distance = d(C) N can be made arbitrarily close to one for suciently large q [15]. Moreover, a polynomial time list decoding algorithm of [9] enables us to nd a nearest codeword x to a given point y provided that s(y; x) N(N D) Choosing D suciently large, we obtain a sequence of ecient t i.p.p. codes with the traceability property (After the draft version of this paper ....

....version of this paper [5] was completed, we became aware of the paper [12] that works out the details of this idea. However, to construct codes of large size following this approach, we have to employ codes over an alphabet of a fairly large size. Namely, recall that by the Plotkin upper bound [15] for any q ary (n; M) code with distance d, M q 1 q : Substituting d n(1 1=t ) we see that for q t the denominator is positive, and therefore, the size of the code M nO t;q (1) Thus, for small q it is not possible to construct codes whose rate R remains bounded away from ....

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M. Tsfasman and S. Vladut, Algebraic-geometric codes, Kluwer, Dordrecht, 1991.

.... Arora and Lund [4] Our methods adapt the proof of the nonapproximability of the shortest lattice vector problem (SVP) due to Micciancio [15] which in turn is based on Ajtai s proof of the hardness of SVP [3] 1 For square prime powers q 49, linear AG codes can perform better than random ones [16] and are constructible in polynomial time. For all other q 46 it is still possible to do better than random codes, however the best known procedures to construct them run in exponential time [18] 2 f(n) is quasi polynomial in n if it grows slower than 2 log c n for some constant c. 2 1.2 ....

....run through the maximum possible interval (0; 1 1 q ) In other words, we prove that GapRNC ( q is NP hard for any relative distance. Given a prime power square q 49; we use long algebraic geometry codes A[l; m; d] q meeting the Tsfasman Vl adut Zink (TVZ) bound d l m l p q 1 (see [16]) The generator matrices A 2 F m l q of these codes can be constructed in polynomial time in l: Note that the TVZ bound also exceeds the GV bound for most code rates and therefore allows to obtain sequences of dense 14 codes with xed q. However, the relative distance d=l is upper bounded by ....

M.A. Tsfasman and S.G. Vladut, Algebraic - Geometric Codes. Dordrecht: Kluwer, 1991.

....makes this a fundamental computational problem in coding theory. The problem gains even more significance in light of the fact that long q ary codes chosen at random give the best parameters 1 known for any q 46 1 For squares q 49, linear AG codes can perform better than random ones [14] and are constructed in polynomial time. For any q 46 it is still (in particular, for q = 2) Such a choice is expected to produce a code of large distance, but no efficient methods are known to lower bound the distance of a code produced in this manner. A polynomial time algorithm to compute the ....

M.A. Tsfasman and S.G. Vladuts, Algebraic - Geometric Codes. Dordrecht: Kluwer, 1991.

....of weight distributions, a method to describe and compute weight distributions, and worked out examples for curves of genus two and three. 1. Introduction The problem that motivated this work comes from coding theory. We sketch the problem in a geometric setting (a setting encouraged in [26]) For a given set of n rational points in projective space over a finite field of q elements, determine the number of hyperplanes that intersect the set in a given number of points. The solution is described by an n 1 tuple of integers, called the weight distribution. For weight distributions of ....

....Class Field. 16 IWAN M. DUURSMA 7. Geometric Goppa codes We first give the definitions and some basic results for general linear codes, and will then consider the class of geometric Goppa codes. For details and for the omitted proofs we refer to [18] 28] 4] for coding theory, and to [29] [26], 20] 24] for geometric Goppa codes. A linear code C of length n is a subspace of the space of all n letter words over a finite field. The elements of C are called codewords. For applications, such as in communication or in information storage, it is important that the codeword as a whole can ....

M.A. Tsfasman and S.G. Vladut. Algebraic-geometric codes. Kluwer Acad. Publ., Dordrecht, The Netherlands, 1990.

....(i.e. submodules of A n , A a ring) via the Gray mapping, which we recall below. In a dierent vein, over the last decade there has been a lot of interest in linear codes coming from algebraic curves over nite elds. The construction of such codes was rst proposed by Goppa in [5] see [15] or [16] for instance. In [17] it is proven that for q 49 a square, there exist sequences of codes over the nite eld with q elements which give asymptotically the best known linear codes over these elds. The second author has extended Goppa s construction to curves over local Artinian rings and shown, ....

M. A. Tsfasman and S. G. Vladut, Algebraic-geometric codes, Kluwer, Dordrecht, 1991.

.... NP = RP (i.e. every problem in NP has a polynomial time probabilistic algorithm that always reject No instances and accepts Yes instances with high probability) Under the stronger assumption that NP does not have random 1 For squares q 49, linear AG codes can perform better than random ones [15] and are constructible in polynomial time. For any q 46 it is still possible to do better than random codes, however the best known procedures to construct them run in exponential time [17] quasi polynomial time 2 algorithms (RQP) we get that the minimum distance of a code of block length n ....

....l = maxfn; l 0 ; 4k ffl g, m = R Delta l and r = ff Delta l. Notice that these settings already satisfy the third condition above, i.e. l n, m R Delta l, and r ff Delta l. We then construct in polynomial time in l, a generator matrix A 2 F m Thetal q for an algebraic geometry code (see [15]) satisfying Delta(C A ) l Gamma m Gamma l p q Gamma1 . Notice that l Gamma m Gamma l p q Gamma 1 = l(1 Gamma R Gamma 1 p q Gamma 1 ) l( 1 2 ffl 2 ) l 2 (1 ffl) r(1 ffl) thus satisfying the first condition. It remains to verify the second condition, namely, ....

M.A. Tsfasman and S.G. Vladuts, Algebraic - Geometric Codes. Dordrecht: Kluwer, 1991.

....to construct linear codes over Z 4Z, with the hope that these codes might have good binary images under the Gray map. In [15] a new method of constructing linear codes over Z 4Z is proposed. The idea there is to generalize the construction of algebraic geometric codes over finite fields ( 12] [13]) to allow the use of a local Artin ring to play the role of the finite field. In that paper, the length, dimension (rank) and minimum Hamming distance of these new codes are computed. However, the crucial Lee weight, which is the same as the Hamming weight of their binary images under the Gray ....

....chain of ideals is eventually stable; since any finite ring obviously satisfies this property, GR(p l , m) # W l (F p m) is an Artin ring. LEE WEIGHTS OF Z 4Z CODES FROM ELLIPTIC CURVES 3 Next, we describe how to generalize the construction of algebraic geometric codes over finite fields ([13], 12] to give codes over local Artin rings such as W l (F p m) # GR(p l , m) The set up for the generalized construction is as follows: Let A be a local Artin ring, and let X be a curve defined over A (i.e. a connected irreducible scheme over Spec A which is smooth and of relative dimension ....

M. A. Tsfasman and S. G. Vladut, Algebraic-geometric codes, Kluwer, Dordrecht, 1991.

....This has been achieved by replacing the Reed Solomon code A by a longer code that is inferior to the Singleton bound. The last two parts of this theorem are related to codes from algebraic curves. A very detailed discussion of these results is carried out in the monograph by Tsfasman and Vladut [157]. In particular, a polynomial time construction of codes in part (c) due to Vladut [162] is given in Chapter 4.3 of that book. The order n 30 has been recently reduced to n 17 in Lop ez Jim enez [112] At the time of writing this there is ongoing research towards reducing this complexity to ....

....between them are filled by shortening or puncturing, The bound in part (d) can be further improved by using multilevel concatenations, but this has never been done because of apparently little interest. The best known bounds for asymptotically good polynomially decodable codes are also found in [157]. This book also gives tables of the bounds for q = 2; 4; 16; 64; 256; and lists the asymptotic behaviour of these and many other bounds in coding theory for ffi 0 and ffi 1 Gamma (1=q) Heuristic algorithms of constructing good short codes are addressed in Honkala and Ostergard [83] 2.1.3. ....

M.A. Tsfasman and S. G. Vladut¸, Algebraic-Geometric Codes, Dordrecht : Kluwer (1991).

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M. Tsfasman and S. Vladut, Algebraic-geometric codes, Kluwer, Dordrecht, 1991.

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M. A. Tsfasman and S. G. Vl adut , Algebraic-Geometric Codes. Dordrecht, The Netherlands: Kluwer, 1991.

....words, if they are in the same orbit for the natural action of the semidirect product of (F q ) and S n . It is clear that this gives a natural equivalence relation on the set of [n; k] q codes. An alternative way to describe codes is via the language of projective systems introduced in [21]. A projective system is a (multi)set X of n points in the projective space P over F q . We call X nondegenerate if these n points are not contained in a hyperplane of P . Two projective systems in P are said to be equivalent if there is a projective automorphism of the ambient space P ....

....between algebraic geometry and coding theory, but also facilitates the introduction of linear codes corresponding to projective algebraic varieties de ned over a nite eld. For more details concerning projective systems and a proof of the above mentioned one to one correspondence, we refer to [21] and [22] 3. Grassmann Codes and Schubert Codes Perhaps the most basic example of a projective algebraic variety over F q is the Grassmannian G ;m = G (V ) of dimensional subspaces of an m dimensional vector space V over F q . We have the well known Pl ucker embedding of the Grassmannian ....

M. A. Tsfasman and S. G. Vladut, \Algebraic Geometric Codes", Kluwer, Amsterdam, 1991.

....a sequence of binary linear codes fCn g of length n = qN; N 1 and distance dn = n=2 such that log A dn NE q ( o(N) 1) 2 Proof We will rst construct a sequence of q ary linear (geometric Goppa) codes. Background information on coding theory and geometry of curves can be looked up in [5]. Let X be a (smooth projective absolutely irreducible) curve of genus g over F q , where q 49 is an even power of a prime. Let N = N(X) X(F q ) be the number of F q rational points of X and suppose that X is such that N g( p q 1) e.g. X is a suitable modular curve) The set of F q ....

M. Tsfasman and S. Vladut, Algebraic-geometric codes, Kluwer, Dordrecht, 1991.

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M. A. Tsfafman and S. G. Vladut, Algebraic-Geometric Codes. Dordrecht/Boston/London: Kluwer, 1991.

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M. A. Tsfasman and S. G. Vladut, Algebraic-Geometric Codes, Kluwer, 1991.

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M. A. Tsfasman and S. G. Vladuts, Algebraic-geometric codes, Kluwer Acad. Publ., Dordrecht, 1991.

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M. A. Tsfasman and S. G. Vl adut , Algebraic--Geometric Codes. Dordrecht, The Netherlands: Kluwer, 1991.

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Tsfasman, M.,Vladut, S., Algebraic-geometric codes, Math. and its Appl., Kluwer Acad. Publishers, Dordrecht, The Netherlands, 1991

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Tsfasman, M.A.; Vladut, S.G.: Algebraic-Geometric Codes. Kluwer Acad. Publ., Amsterdam (1991).

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M. A. Tsfasman and S. G. Vl adut (1991). Algebraic Geometric Codes. Mathematics and Its Application. Kluwer Academic Publishers, The Netherlands.

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Tsfasman, M.,Vladut, S., Algebraic-geometric codes, Math. and its Appl., Kluwer Acad. Publishers, Dordrecht, The Netherlands, 1991

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M.A. Tsfasman and S.G. Vladut, "Algebraic-geometric codes," Kluwer Academic Publishers, 1991.

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Tsfasman, M.,Vladut, S., Algebraic-geometric codes, Math. and its Appl., Kluwer Acad. Publishers, Dordrecht, The Netherlands, 1991

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M.A. Tsfasman and S. G. Vladut¸, Algebraic Geometric Codes, Kluwer 1991.

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