Remarks on generalized toric codes

This note presents some new information on how the minimum distance of the generalized toric code corresponding to a fixed set of integer lattice points S⊂R2S⊂R2 varies with the base field. The main results show that in some cases, over sufficiently large fields, the minimum distance of the code corresponding to a set S   will be the same as that of the code corresponding to the convex hull conv(S)conv(S). In an example, we will also discuss a [49,12,28][49,12,28] generalized toric code over F8F8, better than any previously known code according to M. Grassl's online tables, as of December 2012.