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). The rate of any linear AG code satisfies \begin{align} \frac{k}{n} \geq 1 - \frac{d}{n} - \frac{1}{\sqrt{q}-1}~, \tag*{(1)}\end{align} which comes 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 [11–13].
- Evaluation code— Evaluation codes on varieties can also be considered AG codes since they use algebraic geometry in quantifying code bounds. However, early AG constructions all used only 1D varieties, i.e., algebraic curves.
- Tsfasman-Vladut-Zink (TVZ) code— TVZ codes exceed the GV bound [5].
- EA Galois-qudit stabilizer code— Certain AG codes can be used to construct EA Galois-qudit stabilizer codes [14].
Primary Hierarchy
Parents
Algebraic-geometry (AG) code
Children
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, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), 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, 2022.
- [12]
- M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
- [13]
- I. N. Landjev, “Linear codes over finite fields and finite projective geometries”, Discrete Mathematics 213, 211 (2000) DOI
- [14]
- 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.