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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.

Cousins

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Evaluation code
Evaluation AG codeAG ECC
Residue AG codeAG ECC
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 codeEvaluation Linear \(q\)-ary AG ECC
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.
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]
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
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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} }
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Permanent link:
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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.