Tsfasman-Vladut-Zink (TVZ) code[1]
Description
Member of a family of residue AG or, more generally, evaluation AG codes where \(\cal X\) is either Drinfeld modular curve, a classic modular curve, or a Garcia-Stichtenoth curve.
Rate
TVZ codes, either in the residue AG or evaluation AG constructions, exceed the asymptotic GV bound [1] (see also Ref. [2]). Roughly speaking, this breakthrough result implies that AG codes can outperform random codes.
Parents
- Residue AG code — TVZ codes are evaluation AG codes where \(\cal X\) is either a Drinfeld modular curve, a classic modular curve, or a Garcia-Stichtenoth curve, but can also be formulated as residue AG codes.
- Evaluation AG code — TVZ codes are evaluation AG codes where \(\cal X\) is either a Drinfeld modular curve, a classic modular curve, or a Garcia-Stichtenoth curve, but can also be formulated as residue AG codes.
Cousins
- Algebraic-geometry (AG) code — TVZ codes exceed the GV bound [1].
- Quantum AG code — The AG codes used in an asymptotically good construction of quantum AG codes with non-Clifford transversal gates [3] are those of the TVZ codes.
References
- [1]
- 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
- [2]
- Y. Ihara. "Some remarks on the number of rational points of algebraic curves over finite fields." J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28:721-724 (1982),1981.
- [3]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
Page edit log
- Victor V. Albert (2022-08-05) — most recent
Cite as:
“Tsfasman-Vladut-Zink (TVZ) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/shimura