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| Algebraic geometry codes | |
|---|---|
| Name | Algebraic geometry codes |
| Other names | AG codes |
| Field | Algebraic geometry |
| Introduced | 1981 |
| Developers | Vladimir Drinfeld; Valerii Goppa |
| Notable examples | Goppa code; Hermitian curve codes; Reed–Solomon code |
| Applications | Deep Space Network; Global Positioning System; Cryptography |
Algebraic geometry codes are a class of linear error-correcting codes derived from algebraic curves over finite fields, exploiting the interplay between algebraic geometry and coding theory to produce codes with parameters that can exceed classical bounds. Introduced in work building on ideas of Valerii Goppa and catalyzed by breakthroughs linked to the Tsfasman–Vladut–Zink bound, these codes connect concepts from function field theory, Riemann–Roch theorem, and explicit curves such as the Hermitian curve and modular curves arising from Shimura variety constructions. Their study involves contributions from researchers associated with institutions like the Steklov Institute of Mathematics and the Institut des Hautes Études Scientifiques.
Algebraic geometry codes arise by evaluating functions from the function field of a smooth projective algebraic curve at a set of rational points; this evaluation approach generalizes the evaluation viewpoint of Reed–Solomon codes while importing tools from Riemann–Roch theorem and the arithmetic of curves studied by figures connected to the Weil conjectures and André Weil. The development of asymptotically good families was influenced by results in the theory of modular and Drinfeld modular curves linked to Galois representations and the work surrounding the Tsfasman–Vladut–Zink bound. Prominent examples include constructions based on the Hermitian curve and towers of function fields introduced by authors associated with the University of Bordeaux and Technion – Israel Institute of Technology.
The rigorous construction requires concepts from the theory of function fields over finite fields and the divisor theory developed in classical algebraic geometry texts influenced by authors at the École Normale Supérieure and University of Göttingen. Central ingredients are smooth projective curves, rational points over finite fields such as GF(p)-extensions, divisor classes, and the Riemann–Roch theorem which computes dimensions of spaces of global sections; these tools were refined in the context of arithmetic geometry by contributors associated with the Institute for Advanced Study and results related to the Weil conjectures. The interplay with Galois theory and places of function fields draws on methods used in the study of modular curves and Drinfeld modules.
Given a smooth projective curve C defined over a finite field and a set of distinct rational points P1,...,Pn disjoint from a chosen divisor G, one defines the code by evaluating functions in the Riemann–Roch space L(G) at the Pi, mirroring approaches used in Reed–Solomon code constructions associated with evaluation on the projective line studied in classical works from Princeton University and Harvard University. The choice of curve can come from explicit families like the Hermitian curve or towers produced by authors linked to the Universität Karlsruhe and Universidad Nacional Autónoma de México. The divisor G is chosen to control the dimension via the Riemann–Roch theorem and the minimum distance via pole order considerations similar to techniques discussed in seminars at the Courant Institute of Mathematical Sciences.
Parameters (n, k, d) derive from the number n of rational points, the Riemann–Roch dimension l(G), and the degree of G, with lower bounds on d coming from the Goppa bound; these relationships were central in results leading to the Tsfasman–Vladut–Zink bound that outperforms the Gilbert–Varshamov bound for certain alphabets, a discovery highlighted in conferences at the International Congress of Mathematicians and institutions like the Russian Academy of Sciences. Asymptotically good sequences use towers of function fields with many rational places relative to genus, a theme present in the construction of optimal towers by researchers from ETH Zurich and École Polytechnique.
Decoding methods exploit algebraic structure: syndrome-based approaches generalize those for BCH codes and Reed–Solomon codes; algorithms utilize interpolation and module-theoretic methods reminiscent of the Berlekamp–Massey algorithm and the Sudan algorithm developed in collaborations associated with the University of Waterloo and Massachusetts Institute of Technology. Efficient list-decoding and bounded-distance decoding have been advanced by work leveraging the geometry of the underlying curve, with implementations and complexity analyses presented in venues such as the IEEE International Symposium on Information Theory and workshops hosted by the Institute of Electrical and Electronics Engineers.
Notable families include codes from the Hermitian curve, Garcia–Stichtenoth towers introduced by authors affiliated with the Universidad de Sevilla and Technion – Israel Institute of Technology, and codes from modular and Drinfeld modular curves studied by researchers at the Max Planck Institute for Mathematics and Columbia University. Classical examples relate to Goppa codes and extend Reed–Solomon code theory, while recent explicit constructions draw on work associated with the Clay Mathematics Institute and the Simons Foundation.
Applications span reliable communication in systems like the Deep Space Network and positioning technologies related to the Global Positioning System, and theoretical links reach into cryptography where algebraic geometry inputs inform protocol design examined at the National Institute of Standards and Technology and in industry research at IBM and Google. The subject connects to arithmetic geometry topics such as Weil pairing, Jacobians of curves, and moduli space problems investigated at the Mathematical Sciences Research Institute and within the broader context of research networks including the European Mathematical Society.