Algebraic-geometry (AG) code

Algebraic-geometry (AG) code[1–3]

Description

Binary or \(q\)-ary code or subcode constructed from an algebraic curve of some genus over a finite field via the evaluation construction, the residue construction, or more general constructions that yield nonlinear codes [4; Defs. 15.3.1 and 15.3.2][4; Sec. 15.4.4]. Linear AG codes from the first two constructions are also called geometric Goppa codes.

In alternative conventions (not used here), AG codes are restricted to be linear and/or include evaluation codes defined using algebraic varieties more general than curves.

Rate

Several sequences of linear AG codes beat the GV bound and/or are asymptotically good [5–7][4; Thm. 15.4.3] (see Ref. [8] for details). For square \(q\), there exist sequences of linear AG codes whose asymptotic rate and relative distance satisfy \begin{align} \frac{k}{n} \geq 1 - \frac{d}{n} - \frac{1}{\sqrt{q}-1}~, \tag*{(1)}\end{align} which follows from the Drinfeld-Vladut bound [9]. Nonlinear AG codes can outperform this bound [4; Thm. 15.4.6].

Notes

See book by Goppa [10].

Cousins

Maximum distance separable (MDS) code— Near MDS \([n,k,d]_{p^m}\) AG codes exist when \(n,p,m\) satisfy certain relations according to the Tsfasman-Vladut bound [11–14].
EA Galois-qudit stabilizer code— Certain AG codes can be used to construct EA Galois-qudit stabilizer codes [15].

Member of code lists

Primary Hierarchy

Parents

Algebraic-geometry (AG) code

Children

Evaluation codeGRS GRM

Evaluation codes on algebraic varieties are AG codes. The AG-code literature has mostly focused on codes on algebraic curves, i.e., one-dimensional varieties.

Nonlinear AG code

Cartier code

Every Cartier code is contained in a subfield subcode of a residue AG code. Cartier codes share similar asymptotic properties to subfield subcodes of residue AG codes, with both families admitting sequences of codes that achieve the GV bound.

References

[1]: V. D. Goppa, “Codes Associated with Divisors”, Problemy Peredachi Informatsii, 13:1 (1977), 33–39; Problems of Information Transmission, 13:1 (1977), 22–27
[2]: V. D. Goppa, “Codes on algebraic curves”, Doklady Akademii Nauk SSSR, 259:6 (1981), 1289–1290
[3]: V. D. Goppa, “Algebraico-geometric codes”, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, 46:4 (1982), 762–781; Izvestiya Mathematics, 21:1 (1983), 75–91
[4]: A. Couvreur, H. Randriambololona, “Algebraic Geometry Codes and Some Applications.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[5]: M. A. Tsfasman, S. G. Vlădutx, and Th. Zink, “Modular curves, Shimura curves, and Goppa codes, better than Varshamov‐Gilbert bound”, Mathematische Nachrichten 109, 21 (1982) DOI
[6]: A. Garcia and H. Stichtenoth, “A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound”, Inventiones Mathematicae 121, 211 (1995) DOI
[7]: A. Garcia and H. Stichtenoth, “On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields”, Journal of Number Theory 61, 248 (1996) DOI
[8]: T. Høholdt, J. H. van Lint, and R. Pellikaan, “Algebraic geometry codes,” in Handbook of Coding Theory, Vol. I, Part 1, eds. V. S. Pless and W. C. Huffman (Elsevier, 1998), pp. 871-961
[9]: S. G. Vlăduţ and V. G. Drinfeld, “Number of points of an algebraic curve”, Funktsional’nyi Analiz i ego Prilozheniya, 17:1 (1983), 68–69; Functional Analysis and Its Applications, 17:1 (1983), 53–54
[10]: V. D. Goppa, Geometry and Codes (Springer Netherlands, 1988) DOI
[11]: M. Tsfasman, S. Vlǎduţ, and D. Nogin, Algebraic Geometric Codes: Basic Notions, vol. 139 (American Mathematical Society, 2007)
[12]: M. Tsfasman, S. Vlǎduţ, and D. Nogin, Algebraic Geometric Codes: Advanced Chapters, vol. 238 (American Mathematical Society, 2019)
[13]: M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[14]: I. N. Landjev, “Linear codes over finite fields and finite projective geometries”, Discrete Mathematics 213, 211 (2000) DOI
[15]: L. Sok, “On linear codes with one-dimensional Euclidean hull and their applications to EAQECCs”, (2021) arXiv:2101.06461

Page edit log

Alexander Barg (2024-07-23) — most recent
Victor V. Albert (2024-07-23)
Victor V. Albert (2022-08-09)
Victor V. Albert (2022-03-21)

Cite as:

“Algebraic-geometry (AG) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/ag

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/ag/ag.yml.