Abstract
The well-known Tsfasman-Vladut-Zink (TVZ) theorem states that for all prime powers q = l 2 ≥ 49 there exist sequences of linear codes over \({\mathbb{F}_q}\) with increasing length whose limit parameters R and δ (rate and relative minimum distance) are better than the Gilbert-Varshamov bound. The basic ingredients in the proof of the TVZ theorem are sequences of modular curves (or their corresponding function fields) having many rational points in comparison to their genus (more precisely, these curves attain the so-called Drinfeld-Vladut bound). Starting with such a sequence of curves and using Goppa’s construction of algebraic geometry (AG) codes, one easily obtains sequences of linear codes whose limit parameters beat the Gilbert-Varshamov bound.
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References
Stichtenoth, H.: Transitive and Self-Dual Codes Attaining the Tsfasman-Vladut-Zink Bound. IEEE Trans. Inform. Theory 52, 2218–2224 (2006)
Bassa, A., Garcia, A., Stichtenoth, H.: A New Tower over Cubic Finite Fields (preprint, 2007)
Bassa, A., Stichtenoth, H.: Asymptotic Bounds for Transitive and Self-Dual Codes over Cubic Finite Fields (in preparation, 2007)
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Stichtenoth, H. (2007). Nice Codes from Nice Curves. In: Boztaş, S., Lu, HF.(. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2007. Lecture Notes in Computer Science, vol 4851. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77224-8_8
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DOI: https://doi.org/10.1007/978-3-540-77224-8_8
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