Computing Vanishing Ideals for Toric Codes
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Motivated by applications to the theory of error-correcting codes, we give an algorithmic method for computing a generating set for the ideal generated by β-graded polynomials vanishing on a subset of a simplicial complete toric variety X over a finite field 𝔽_q, parameterized by rational functions, where β is a d× r matrix whose columns generate a subsemigroup ℕβ of ℕ^d. We also give a method for computing the vanishing ideal of the set of 𝔽_q-rational points of X. When β=[w_1 ⋯ w_r] is a row matrix corresponding to a numerical semigroup ℕβ=⟨ w_1,…,w_r ⟩, X is a weighted projective space and generators of its vanishing ideal is given using generators of defining (toric) ideals of numerical semigroup rings corresponding to semigroups generated by subsets of {w_1,…,w_r}.
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