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arXiv · 1512.06868

Minimum distance functions of graded ideals and Reed-Muller-type codes

Jose Martinez-Bernal·Yuriko Pitones·Rafael H. Villarreal
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Abstract

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives an algebraic formulation of the minimum distance of a projective Reed-Muller-type code in terms of the algebraic invariants and structure of the underlying vanishing ideal. Then we give a method, based on Groebner bases and Hilbert functions, to find lower bounds for the minimum distance of certain Reed-Muller-type codes. Finally we show explicit upper bounds for the number of zeros of polynomials in a projective nested cartesian set and give some support to a conjecture of Carvalho, Lopez-Neumann and Lopez.

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Jose Martinez-Bernal, Yuriko Pitones, Rafael H. Villarreal. 2016-06-12. Minimum distance functions of graded ideals and Reed-Muller-type codes. https://arxiv.org/abs/1512.06868

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