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Journal Articles International Journal of Number Theory Year : 2021

Projective toric codes

Abstract

Any integral convex polytope $P$ in $\mathbb{R}^N$ 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 $\mathcal{L}(D_P)$ 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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Dates and versions

hal-03142469 , version 1 (28-11-2023)

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Cite

Jade Nardi. Projective toric codes. International Journal of Number Theory, 2021, 18 (01), pp.179-204. ⟨10.1142/S1793042122500142⟩. ⟨hal-03142469⟩
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