Projective toric codes

03/23/2020

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by   Jade Nardi, et al.

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Any integral convex polytope P in R^N provides a 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 X_P , obtained by evaluating global section of L(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 non-zero 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 lowerbound on the minimum distance is described.