Projective toric codes

Any integral convex polytope P in R-N provides an N-dimensional toric variety X-P and an ample divisor D-P on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on XP, obtained by evaluating global section of the line bundle corresponding to D-P on every rational point of X-P. This work presents an extension of toric codes analogous to the one of Reed-Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope P and an algorithmic technique to get a lower bound on the minimum distance is described.