Codes and algebraic curves
Some Properties of Elliptic Codes Over a Field of Characteristic 2
AAECC-3 Proceedings of the 3rd International Conference on Algebraic Algorithms and Error-Correcting Codes
An Extension of Kedlaya's Algorithm to Artin-Schreier Curves in Characteristic 2
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Vector bundles and codes on algebraic curves
Vector bundles and codes on algebraic curves
A geometric view of decoding AG codes
AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
On constructing AG codes without basis functions for riemann-roch spaces
AAECC'06 Proceedings of the 16th international conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Construction and decoding of a class of algebraic geometry codes
IEEE Transactions on Information Theory
On the decoding of algebraic-geometric codes
IEEE Transactions on Information Theory
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Motivated by error-correcting coding theory, we pose some hard questions regarding moduli spaces of rank-2 vector bundles over algebraic curves. We propose a new approach to the role of rank-2 bundles in coding theory, using recent results over the complex numbers, namely restriction of vector bundles from the projective space where the curve is embedded. We specialize our analysis to plane quartic curves which, if smooth, are canonical curves of genus three, and remark that all the bundles in question are restrictions. Using the vector-bundle approach, we work out explicit equations for the error divisors viewed as points of a multisecant variety. We specialize canonical quartics even more, to Klein's curve, and finite fields of characteristic two, a situation in which bundles can be neatly trivialized and codes have been produced. We give explicit equations, work out counting results for curves, Jacobians, and varieties of bundles, revealing several surprising features.