[Jump to code hierarchy]

Tsfasman-Vladut-Zink (TVZ) code[1]

Description

Member of a family of AG codes obtained from algebraic curves via the residue or evaluation construction. Sequences of curves with many rational points, such as Drinfeld modular curves, classical modular curves, or Garcia-Stichtenoth curves, yield the asymptotic parameters of the TVZ bound [2; Sec. 15.4.2].

Rate

TVZ codes, either in the residue AG or evaluation AG constructions, exceed the asymptotic GV bound [1] (see also Ref. [3]). For square \(q \geq 49\), these codes improve on the GV bound; for smaller \(q\), they do not [2; Sec. 15.4.2]. Roughly speaking, this result implies that AG codes can outperform random codes.

Cousin

  • Quantum AG code— The AG codes used in an asymptotically good construction of quantum AG codes with non-Clifford transversal gates [4] are those of the TVZ codes.

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Evaluation codeAG ECC
Evaluation AG code
Residue AG code
Parents
Residue AG code
TVZ codes can be formulated as residue AG codes on algebraic curves.
Evaluation AG codeEvaluation Linear \(q\)-ary AG ECC
TVZ codes can also be formulated as evaluation AG codes on algebraic curves; these are dual to the corresponding residue AG codes.
Tsfasman-Vladut-Zink (TVZ) code

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]
A. Couvreur, H. Randriambololona, “Algebraic Geometry Codes and Some Applications.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[3]
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
[4]
L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: shimura

Cite as:
“Tsfasman-Vladut-Zink (TVZ) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/shimura
BibTeX:
@incollection{eczoo_shimura, title={Tsfasman-Vladut-Zink (TVZ) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/shimura} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/shimura

Cite as:

“Tsfasman-Vladut-Zink (TVZ) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/shimura

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