Abstract
Toric codes are examples of evaluation codes. They are produced by evaluating homogeous polynomials of a fixed degree at the \({\mathbb {F}}_q\)-rational points of a subset Y of a toric variety X. These codes reveal how algebraic geometry and coding theory are interrelated. The minimum distance of a code is the minimum number of nonzero entries in the codewords of the code. Let I(Y) be the ideal generated by all homogeneous polynomials vanishing at all the points of Y, which is also known as the vanishing ideal of Y. We give three algebraic algorithms computing the minimum distance by using commutative algebraic tools such as the multigraded Hilbert polynomials of ideals obtained from I(Y) and zero divisors f of I(Y), and primary decomposition of I(Y), for finding a homogeneous polynomial f among all homogeneous polynomials of the same degree which has the maximum number of roots on Y.
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Acknowledgements
Fadime Baldemir is supported by TÜBİTAK (TÜBİTAK-2211-A National PhD Scholarship Program). Mesut Şahin is supported by TÜBİTAK Project No:119F177. This article is part of Fadime Baldemir’s PhD thesis.
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Baldemir, F., Şahin, M. Calculating the Minimum Distance of a Toric Code via Algebraic Algorithms. Math.Comput.Sci. 17, 20 (2023). https://doi.org/10.1007/s11786-023-00566-7
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DOI: https://doi.org/10.1007/s11786-023-00566-7