AI 中文总结
本文对Tohǎneanu与Van Tuyl关于约化完全交点码最小距离的猜想给出肯定解答，通过超定多项式方程组Bézout界的改进形式证明，并推广得到此类码广义汉明重量的界。
AI 中文摘要
针对由约化完全交点集构造的码的最小距离，我们对Tohǎneanu和Van Tuyl提出的猜想给出了肯定解答。尽管该猜想具有技术性，我们证明它可直接从超定多项式方程组经典Bézout界的一个鲜为人知的改进形式导出，为完整起见，本文给出该改进界的自包含证明。此外，我们表明采用相同方法可得到此类码的广义汉明重量界，更一般地，可控制零维完全交点上d次形式求值所得码的最小距离。
英文摘要
We provide a positive answer to a conjecture proposed by Tohǎneanu 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 Bézout 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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