Decoding by rank-2 bundles over plane quartics

Abstract

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.