Projective toric codes

Abstract

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. 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 [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.