A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound

Garcia, Arnaldo; Stichtenoth, Henning

doi:10.1007/bf01884295

Cited by 296 publications

(295 citation statements)

References 8 publications

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“…There are a few AG codes in existence today (see [26,18,16]). We choose for sake of clarity (as did [20]) to focus on the family of codes given by the explicit and fairly simply Garcia-Stichtenoth function fields [16]. More details are given in Section 6.…”

Section: Definition 4 (Ag Code)mentioning

confidence: 99%

“…Garcia and Stichtenoth [16] give an explicit construction of function fields F n extending F q 2 . The main theorem of the paper is that F n has at least s(n) + 1 def = q n−1 (q 2 − 1) + 1 places of degree one and genus at most g(n) def = q n−1 (q + 1).…”

Section: Construction Based On Ag Codesmentioning

confidence: 99%

“…The function fields underlying the work of [26], however, are far from being explicit. The main contribution of [16]-can be seen as giving a simple construction of function fields attaining the Drinfeld-Vladut bound. Indeed, we can describe the construction here: Let F 1 = F q 2 (x 1 ), the rational function field over one variable x 1 .…”

Section: Construction Based On Ag Codesmentioning

confidence: 99%

See 2 more Smart Citations

Coding Constructions for Blacklisting Problems without Computational Assumptions

Kumar

Rajagopalan

Sahai³

1999

Advances in Cryptology — CRYPTO’ 99

127

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Abstract. We consider the broadcast exclusion problem: how to transmit a message over a broadcast channel shared by N = 2 n users so that all but some specified coalition of k excluded users can understand the contents of the message. Using error-correcting codes, and avoiding any computational assumptions in our constructions, we construct natural schemes that completely avoid any dependence on n in the transmission overhead. Specifically, we construct: (i) (for illustrative purposes,) a randomized scheme where the server's storage is exponential (in n), but the transmission overhead is O(k), and each user's storage is O(kn); (ii) a scheme based on polynomials where the transmission overhead is O(kn) and each user's storage is O(kn); and (iii) a scheme using algebraic-geometric codes where the transmission overhead is O(k 2 ) and each user is required to store O(kn) keys. In the process of proving these results, we show how to construct very good cover-free set systems and combinatorial designs based on algebraic-geometric codes, which may be of independent interest and application. Our approach also naturally extends to solve the problem in the case where the broadcast channel may introduce errors or lose information.

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Section: Definition 4 (Ag Code)mentioning

confidence: 99%

Section: Construction Based On Ag Codesmentioning

confidence: 99%

Section: Construction Based On Ag Codesmentioning

confidence: 99%

See 1 more Smart Citation

Coding Constructions for Blacklisting Problems without Computational Assumptions

Kumar

Rajagopalan

Sahai³

1999

Advances in Cryptology — CRYPTO’ 99

127

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“…Since the error probability of our algorithm does not increase with n, it is possible to consider long codes over the alphabet F q . In particular, using codes from asymptotically optimal curves over F q 2 [7,4], and assuming that m is large enough, our codes will be able to reliably decode over a Q-ary symmetric channel with error probability p, and maintain a rate close to…”

Section: Introductionmentioning

confidence: 99%

Improved Decoding of Interleaved AG Codes

Brown

Minder

Shokrollahi

2005

Lecture Notes in Computer Science

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Abstract. We analyze a generalization of a recent algorithm of Bleichenbacher et al. for decoding interleaved codes on the Q-ary symmetric channel for large Q. We will show that for any m and any the new algorithms can decode up to a fraction of at least βm βm+1 (1−R−2Q −1/2m )− errors, where β = ln(q m −1) ln(q m ) , and that the error probability of the decoder is upper bounded by O(1/q n ), where n is the block-length. The codes we construct do not have a-priori any bound on their length.

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“…So the computation of the integral closure will be done here by invoking the q-th power algorithm introduced by Leonard [10], using the above monomial ordering to define normal forms, and using a variant of the trace-dual basis of the standard basis to define the initial set ∆ * 0 in the algorithm. This algorithm is used to compute missing functions (see Pellikaan [13] and an example in Leonard [11]) for the AG codes from the towers of function fields introduced by Garcia and Stichtenoth [5] and [6]; that is (slightly more generally) if U c is the set of points at which at least one element of R is not regular, then the algorithm computes the set of functions regular on U . But it can be viewed as an algorithm for producing the integral closure of a given ring or the normalization or non-singular model of a curve, particularly one in special position.…”

Section: Introductionmentioning

confidence: 99%

Integral closures and weight functions over finite fields

Leonard

Pellikaan

Proceedings IEEE International Symposium on Information Theory,

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Curves and surfaces of type I are generalized to integral towers of rank r. Weight functions with values in N r and the corresponding weighted total-degree monomial orderings lift naturally from one domain R j−1 in the tower to the next, R j , the integral closure of R j−1 [x j ]/ φ(x j ) . The q-th power algorithm is reworked in this more general setting to produce this integral closure over finite fields, though the application is primarily that of calculating the normalizations of curves related to one-point AG codes arising from towers of function fields. Every attempt has been made to couch all the theory in terms of multivariate polynomial rings and ideals instead of the terminology from algebraic geometry or function field theory, and to avoid the use of any type of series expansion.

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A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound

Cited by 296 publications

References 8 publications

Coding Constructions for Blacklisting Problems without Computational Assumptions

Coding Constructions for Blacklisting Problems without Computational Assumptions

Improved Decoding of Interleaved AG Codes

Integral closures and weight functions over finite fields

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