Toric surface codes and Minkowski length of polygons

In this paper we prove new lower bounds for the minimum distance of a toric surface code defined by a convex lattice polygon P. The bounds involve a geometric invariant L(P), called the full Minkowski length of P which can be easily computed for any given P.Comment: 18 pages, 9 figure

Asymptotically maximal families of hypersurfaces in toric varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $\Delta$ there are families of hypersurfaces with the Newton polytopes $(\lambda\Delta)_{\lambda \in \mathbb{N}}$ that are asymptotically maximal when $\lambda$ tends to infinity. We also show that these results generalize to complete intersections.Comment: 18 pages, 1 figur

Curves of every genus with many points, I: Abelian and toric families

Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs they constructed curves with many points in infinitely many genera; however, their sequences of genera are somewhat sparse. In this paper, we prove that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we use curves on toric surfaces to prove that liminf_{g-->infty} N_q(g)/g^{1/3} > 0; we also show that these results are the best possible that can be proved with these families of curves.Comment: LaTeX, 20 page