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

Description

Member of a family of residue AG codes where \(\cal X\) is either a reduction of a Shimura curve or an elliptic curve of varying genus.

Rate

TVZ codes exceed the asymptotic Gilbert-Varshamov (GV) bound [1] (see also Ref. [2]). Roughly speaking, this breakthrough result implies that AG codes can outperform random codes. Such families of codes are optimal.

Parent

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.
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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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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/tree/main/codes/classical/q-ary_digits/ag/residueAG/shimura.yml.