A problem of Beelen, Garcia and Stichtenoth on an Artin–Schreier tower in characteristic two

Chara, María; Navarro, Horacio; Toledano, R.

doi:10.4064/aa8638-10-2017

Cited by 3 publications

(6 citation statements)

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“…In this note we have completed the classification of the asymptotic behavior of the tower H over a finite field with 2 s in terms of the parity of s. This work was started by (Chara et al, 2018). As stated by (Beelen et al, 2006) there are only four recursive towers of Artin-Schreier type over a field with two elements including the tower H .…”

Section: Discussionmentioning

confidence: 98%

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On the splitting rate of rate of a tower of Artin-Schreier type

Navarro

2020

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat.

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Section: Discussionmentioning

confidence: 98%

“…In (Chara et al, 2018) was proven that sequence H = {F i } ∞ i=0 of function fields defined by the Artin-Schreier equation…”

Section: Introductionmentioning

confidence: 99%

On the splitting rate of rate of a tower of Artin-Schreier type

Navarro

2020

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat.

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“…In this subsection, we would like to give two further examples of optimal, recursive towers. The first one is taken from [49], the second one from [22]. E.C.…”

Section: Two Further Examples Of Optimal Recursive Towersmentioning

confidence: 99%

“…A second, more recently found optimal tower over F 4 given in [22], is defined recursively by the equation…”

Section: Two Further Examples Of Optimal Recursive Towersmentioning

confidence: 99%

A survey on recursive towers and Ihara's constant

Beelen¹

2022

Preprint

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Since Serre gave his famous Harvard lectures in 1985 on various aspects of the theory of algebraic curves defined over a finite field, there have been many developments. In this survey article, an overview will be given on the developments concerning the quantity A(q), known as Ihara's constant. The main focus will be on explicit techniques and in particular recursively defined towers of function fields over a finite field, which have given good lower bounds for Ihara's constant in the past.

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“…In our opinion, the similarity of settings between foldable codes and (asymptotically good) towers of curves makes the latter one of the most promising candidates for new foldable codes. If one could assert that a tower (for instance of Kummer extensions [NOQ12] or of Artin-Schreier type [GS95,CNT18]) comes with a sequence of compatible divisors, this would provide a sequence of foldable codes.…”

Section: Towards Other Foldable Ag Codes Familiesmentioning

confidence: 99%

Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

Bordage¹,

Nardi²

2020

Preprint

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In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code C = C(C, P, D) is a vector space associated to evaluations on P of functions in the Riemann-Roch space L C (D). The problem of testing proximity to an error-correcting code C consists in distinguishing between the case where an input word, given as an oracle, belongs to C and the one where it is far from every codeword of C. AG codes are good candidates to construct short proof systems, but there exists no efficient proximity tests for them. We aim to fill this gap.We construct an Interactive Oracle Proof of Proximity (IOPP) for some families of AG codes by generalizing an IOPP for Reed-Solomon codes, known as the FRI protocol [BBHR18]. We identify suitable requirements for designing efficient IOPP systems for AG codes. In addition to proposing the first proximity test targeting AG codes, our IOPP admits quasilinear prover arithmetic complexity and sublinear verifier arithmetic complexity with constant soundness for meaningful classes of AG codes. We take advantage of the algebraic geometry framework that makes any group action on the curve that fixes the divisor D translate into a decomposition of the code C. Concretely, our approach relies on Kani's result that splits the Riemann-Roch space of any invariant divisor under this action into several explicit Riemann-Roch spaces on the quotient curve [Kan86]. Under some hypotheses, these spaces behave well enough to define an AG code C ′ on the quotient curve so that a proximity test to C can be reduced to one to C ′ . Iterating this process thoroughly, we end up with a membership test to a code with significantly smaller length.

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A problem of Beelen, Garcia and Stichtenoth on an Artin–Schreier tower in characteristic two

Cited by 3 publications

References 0 publications

On the splitting rate of rate of a tower of Artin-Schreier type

On the splitting rate of rate of a tower of Artin-Schreier type

A survey on recursive towers and Ihara's constant

Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

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