A low complexity algorithm for the construction of algebraic geometric codes better than the Gilbert-Varshamov bound

Shum, Kenneth W.; Kumar, P. Vijay; Aleshnikov, I.; Deolalikar, Vinay

doi:10.1109/isit.2001.936173

Cited by 37 publications

(75 citation statements)

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“…In [3,5,6] polynomial-time algorithms are given for some specific towers. We believe that it is possible to provide such a polynomial time algorithm also for the tower T , and hence to obtain an explicit construction of the codes C i in polynomial time.…”

Section: Remark A19mentioning

confidence: 99%

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Ben–Sasson

Kaplan

Kopparty

et al. 2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, i.e., how long is the PCP compared to the original NP-proof. The state-of-the-art work of Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)) shows that one can encode proofs of length n by PCPs of length n · poly log n that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to n ε , then one can construct PCPs of length O(n) for circuit-SAT, and PCPs of length O(t log t) for any language in NTIME(t).More specifically, for any ε > 0 we present (non-uniform) probabilistically checkable proofs (PCPs) of length 2 O(1/ε) · n that can be checked using n ε queries for circuit-SAT instances of size n. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity (o(n)).Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed in the appendix to this paper for the first time for every message length, after they have been constructed for infinitely many message lengths by Stichtenoth [Trans. Information Theory 2006].

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Section: Remark A19mentioning

confidence: 99%

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Ben–Sasson

Kaplan

Kopparty

et al. 2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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“…It is an important goal to improve the list size to a constant independent of n. The dependence of the list size on n in [4] arises because Reed-Solomon codes need an alphabet of size at least n. This motivates one to generalize this approach to AG codes which can have arbitrary block lengths over fixed alphabets, and also have very nice algebraic properties. Recent advances have greatly improved the efficiency and explicitness of constructions of AG codes [12], making this a promising route to approach capacity with better list size and decoding complexity.…”

Section: Introductionmentioning

confidence: 99%

“…In [12], a polynomial time algorithm is presented to compute the generator matrix of such an AG code. All we need to add to this to achieve the representation needed in Section 5.3 are the evaluations of the basis elements at some place R of a specified large degree, and the evaluations of 2g extra functions at the code places P 1 , P 2 , .…”

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confidence: 99%

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Correlated Algebraic-Geometric Codes: Improved List Decoding over Bounded Alphabets

Guruswami

Patthak

2006

2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)

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Abstract. We define a new family of error-correcting codes based on algebraic curves over finite fields, and develop efficient list decoding algorithms for them. Our codes extend the class of algebraic-geometric (AG) codes via a (nonobvious) generalization of the approach in the recent breakthrough work of Parvaresh and Vardy (2005).Our work shows that the PV framework applies to fairly general settings by elucidating the key algebraic concepts underlying it. Also, more importantly, AG codes of arbitrary block length exist over fixed alphabets Σ, thus enabling us to establish new trade-offs between the list decoding radius and rate over a bounded alphabet size.The work of Parvaresh and Vardy (2005) was extended in Guruswami and Rudra (2006) to give explicit codes that achieve the list decoding capacity (optimal trade-off between rate and fraction of errors corrected) over large alphabets. A similar extension of this work along the lines of Guruswami and Rudra could have substantial impact. Indeed, it could give better trade-offs than currently known over a fixed alphabet (say, GF(2 12 )), which in turn, upon concatenation with a fixed, well-understood binary code, could take us closer to the list decoding capacity for binary codes. This may also be a promising way to address the significant complexity drawback of the result of Guruswami and Rudra, and to enable approaching capacity with bounded list size independent of the block length (the list size and decoding complexity in their work are both n Ω(1/ε) where ε is the distance to capacity).Similar to algorithms for AG codes from Sudan (1999) and, our encoding/decoding algorithms run in polynomial time assuming a natural polynomial-size representation of the code. For codes based on a specific "optimal" algebraic curve, we also present an expected polynomial time algorithm to construct the requisite representation. This in turn fills an important void in the literature by presenting an efficient construction of the representation often assumed in the list decoding algorithms for AG codes.

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“…But there are two more recent methods [16] and [10] using methods which start with a module containing the integral closure and delete elements not in the integral closure.…”

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 low complexity algorithm for the construction of algebraic geometric codes better than the Gilbert-Varshamov bound

Cited by 37 publications

References 4 publications

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Correlated Algebraic-Geometric Codes: Improved List Decoding over Bounded Alphabets

Integral closures and weight functions over finite fields

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