Toward good families of codes from towers of surfaces

Alain Couvreur; Philippe Lebacque; Marc Perret

doi:10.1090/conm/770/15431

Toward good families of codes from towers of surfaces

Alain Couvreur
, Philippe Lebacque
, Marc Perret

UFR Sciences et techniques
Universite Jean-Jaures

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be “liftable” under finite morphisms, paving the way toward the construction of good codes from towers of surfaces. In the same direction, we establish a criterion for a surface with a fixed finite set of closed points P to have an infinite tower of ℓ–étale covers in which P splits totally. We conclude by stating several open problems. In particular, we relate the existence of asymptotically good codes from general type surfaces with a very ample canonical class to the behaviour of their number of rational points with respect to their K² and coherent Euler characteristic.

Original language	English
Title of host publication	Arithmetic, Geometry, Cryptography and Coding Theory - 17th International Conference Arithmetic, Geometry, Cryptography and Coding Theory, 2019
Editors	Stéphane Ballet, Gaetan Bisson, Irene Bouw
Publisher	American Mathematical Society
Pages	59-101
Number of pages	43
ISBN (Print)	9781470454265
DOIs	https://doi.org/10.1090/conm/770/15431
Publication status	Published - 1 Jan 2021
Event	17th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory, AGC2T-17 2019 - Marseille, France Duration: 10 Jun 2019 → 14 Jun 2019

Publication series

Name	Contemporary Mathematics
Volume	770
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	17th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory, AGC2T-17 2019
Country/Territory	France
City	Marseille
Period	10/06/19 → 14/06/19

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10.1090/conm/770/15431

Cite this

Couvreur, A., Lebacque, P., & Perret, M. (2021). Toward good families of codes from towers of surfaces. In S. Ballet, G. Bisson, & I. Bouw (Eds.), Arithmetic, Geometry, Cryptography and Coding Theory - 17th International Conference Arithmetic, Geometry, Cryptography and Coding Theory, 2019 (pp. 59-101). (Contemporary Mathematics; Vol. 770). American Mathematical Society. https://doi.org/10.1090/conm/770/15431

Couvreur, Alain ; Lebacque, Philippe ; Perret, Marc. / Toward good families of codes from towers of surfaces. Arithmetic, Geometry, Cryptography and Coding Theory - 17th International Conference Arithmetic, Geometry, Cryptography and Coding Theory, 2019. editor / Stéphane Ballet ; Gaetan Bisson ; Irene Bouw. American Mathematical Society, 2021. pp. 59-101 (Contemporary Mathematics).

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title = "Toward good families of codes from towers of surfaces",

abstract = "We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be “liftable” under finite morphisms, paving the way toward the construction of good codes from towers of surfaces. In the same direction, we establish a criterion for a surface with a fixed finite set of closed points P to have an infinite tower of ℓ–{\'e}tale covers in which P splits totally. We conclude by stating several open problems. In particular, we relate the existence of asymptotically good codes from general type surfaces with a very ample canonical class to the behaviour of their number of rational points with respect to their K2 and coherent Euler characteristic.",

author = "Alain Couvreur and Philippe Lebacque and Marc Perret",

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Couvreur, A, Lebacque, P & Perret, M 2021, Toward good families of codes from towers of surfaces. in S Ballet, G Bisson & I Bouw (eds), Arithmetic, Geometry, Cryptography and Coding Theory - 17th International Conference Arithmetic, Geometry, Cryptography and Coding Theory, 2019. Contemporary Mathematics, vol. 770, American Mathematical Society, pp. 59-101, 17th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory, AGC2T-17 2019, Marseille, France, 10/06/19. https://doi.org/10.1090/conm/770/15431

Toward good families of codes from towers of surfaces. / Couvreur, Alain; Lebacque, Philippe; Perret, Marc.
Arithmetic, Geometry, Cryptography and Coding Theory - 17th International Conference Arithmetic, Geometry, Cryptography and Coding Theory, 2019. ed. / Stéphane Ballet; Gaetan Bisson; Irene Bouw. American Mathematical Society, 2021. p. 59-101 (Contemporary Mathematics; Vol. 770).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Perret, Marc

PY - 2021/1/1

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N2 - We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be “liftable” under finite morphisms, paving the way toward the construction of good codes from towers of surfaces. In the same direction, we establish a criterion for a surface with a fixed finite set of closed points P to have an infinite tower of ℓ–étale covers in which P splits totally. We conclude by stating several open problems. In particular, we relate the existence of asymptotically good codes from general type surfaces with a very ample canonical class to the behaviour of their number of rational points with respect to their K2 and coherent Euler characteristic.

AB - We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be “liftable” under finite morphisms, paving the way toward the construction of good codes from towers of surfaces. In the same direction, we establish a criterion for a surface with a fixed finite set of closed points P to have an infinite tower of ℓ–étale covers in which P splits totally. We conclude by stating several open problems. In particular, we relate the existence of asymptotically good codes from general type surfaces with a very ample canonical class to the behaviour of their number of rational points with respect to their K2 and coherent Euler characteristic.

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BT - Arithmetic, Geometry, Cryptography and Coding Theory - 17th International Conference Arithmetic, Geometry, Cryptography and Coding Theory, 2019

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Couvreur A, Lebacque P, Perret M. Toward good families of codes from towers of surfaces. In Ballet S, Bisson G, Bouw I, editors, Arithmetic, Geometry, Cryptography and Coding Theory - 17th International Conference Arithmetic, Geometry, Cryptography and Coding Theory, 2019. American Mathematical Society. 2021. p. 59-101. (Contemporary Mathematics). doi: 10.1090/conm/770/15431

Toward good families of codes from towers of surfaces

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