ON THE MINIMUM DISTANCE OF A TORIC CODE VIA VANISHING IDEAL

Date

2023-8-10

Author

Baldemir, Fadime

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Toric codes are examples of evaluation codes produced by evaluating homogeous polynomials of a fixed degree $\alpha$ at the $\F_q$-rational points of a subset $Y$ of a toric variety $X$. The kernel of the evaluation map is just the subspace $I_{\alpha}(Y)$ generated by the elements of degree $\alpha$. Therefore, it is an important algebraic invariant that determines the parameters of the code. One of the main objective of this thesis is to calculate the minimum distance of an evaluation code obtained from a general smooth toric variety by using $I_{\alpha}(Y)$. Let $I(Y)$ be the ideal generated by all homogeneous polynomials vanishing at all the points of $Y$. We give three algorithms and share a comparative table, discussing the computation time. We use commutative algebraic tools such as the multigraded Hilbert polynomials of ideals derived from $I(Y)$, zero divisors $f$ of $I(Y)$, and primary decomposition of $I(Y)$ to calculate the minimum distance. Their utilization enables us to find a homogeneous polynomial $f$ among all homogeneous polynomials of the same degree which has the maximum number of roots on $Y$. Another primary objective is to focus on the dimension of toric codes obtained from Kleinschmidt toric variety, a generalization of Hirzebruch surfaces. Let $X_{\Sigma_3(l)}$ be a Kleinschmidt toric variety. We compute the dimension of a toric code $\mathcal{C}_{\alpha,Y}$ for any degree $\alpha=(c,d)\in \mathbb N\beta$, for any subgroup $Y=V_{X}(I_L)(\mathbb F_q)$ of the torus $T_{\Sigma_3(l)}$ both algebraically and geometrically by counting the lattice points of the polytope corresponding to $S_\alpha/I_\alpha.$

Subject Keywords

minimum distance, toric code, dimension of a toric code, Kleinschmidt toric variety

URI

https://hdl.handle.net/11511/105102

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Graduate School of Natural and Applied Sciences, Thesis