Scroll codes over curves of higher genus

Abstract

We construct linear codes from scrolls over curves of high genus and study the higher support weights di of these codes. We embed the scroll into projective space P k−1 and calculate bounds for the di by considering the maximal number of Fq-rational points that are contained in a codimension h subspace of P k−1 . We nd lower bounds of the di and for the cases of large i calculate the exact values of the di.This work follows the natural generalisation of Goppa codes to higher-dimensional varieties as studied by

Scroll codes over curves of higher genus: reducible and superstable vector bundles

“…For any linear code C and any integer h ≥ 1, let D h (C) be the set of all linear subspaces of C generated by h linearly independent elements of C. Set codimension h linear subspace of P k−1 . Johnsen and Rasmussen gave lower bounds for the integers d h (C) when T = P(E) with E a semistable vector bundle on a smooth curve [11]. A nice feature of [11] is that these lower bounds are the same for all E on a given curve.…”

“…Johnsen and Rasmussen gave lower bounds for the integers d h (C) when T = P(E) with E a semistable vector bundle on a smooth curve [11]. A nice feature of [11] is that these lower bounds are the same for all E on a given curve. Here we argue that for certain (very easy to construct) vector bundles these lower bounds may be improved and give the missing pieces (to me) to make further improvements.…”

“…Here we argue that for certain (very easy to construct) vector bundles these lower bounds may be improved and give the missing pieces (to me) to make further improvements. A very useful feature of [11] is that the bound in that paper are true for all semistable vector bundles. The bounds in [11] do not depend on F q , in the sense that they only depend from the geometry of the given curve over F q .…”

“…A very useful feature of [11] is that the bound in that paper are true for all semistable vector bundles. The bounds in [11] do not depend on F q , in the sense that they only depend from the geometry of the given curve over F q . As stressed in [22] it should be possible to get better bounds (in some cases) using that the gonality of a curve defined over F q may come from pencils not defined over F q .…”

“…Following [11] and [7] we recall the following construction of certain evaluation codes coming from a projective bundle over a smooth curve. Fix an integer k ≥ 2.…”