Projective toric codes

2020/03/23 by Jade Nardi, Nardi, Jade · 1 citation
Computer Science · Mathematics · #13P10 #14J20 #14M25 #52B05 #52B11 #52B20 #94B05 #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.IT #math.AG #math.IT #msc:13P10 #msc:14J20 #msc:14M25 #msc:52B05 #msc:52B11 #msc:52B20 #msc:94B05

paper · pdf · doi:10.48550/arxiv.2003.10357

arxiv created 2021/02/04 · arxiv updated 2021/02/08

Abstract

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.

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• Maximum number of zeroes of polynomials on weighted projective spaces over a finite field

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