Generalized Hamming weights of codes arising from complete intersection

Eduardo Camps Moreno; Flavio Salizzoni; Rodrigo San-José

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arXiv · 2608.19978

Generalized Hamming weights of codes arising from complete intersection

Eduardo Camps Moreno·Flavio Salizzoni·Rodrigo San-José

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Abstract

We provide a positive answer to a conjecture proposed by Toh\v{a}neanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classical B\'ezout bound for overdetermined polynomial systems. For completeness, this paper presents a self-contained proof of this refined bound. Furthermore, we show that using the same approach, it is possible to obtain a bound on the generalized Hamming weights of such a code and, more generally, to control the minimum distance of the codes obtained by evaluating forms of degree $d$ on the points of a zero-dimensional complete intersection.

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Eduardo Camps Moreno, Flavio Salizzoni, Rodrigo San-José. 2026-08-20. Generalized Hamming weights of codes arising from complete intersection. https://arxiv.org/abs/2608.19978

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