二元Hamming球面的消失理想
Vanishing ideals of binary Hamming spheres
摘要：
我们考虑简化的代数范式（sANF）的布尔函数消失的汉明球面为中心的零和相关的sANF向量。我们表明，这个向量是周期性的，导致一个有效的计算sANF和特定的情况下的具体公式。此外，我们明确地提供了一个连接到二进制M{<$}bius变换的初等对称函数。最后，我们提出了一种基于多项式估计的方法来限制二元非线性码的最小距离。同样的方法可以用来计算二元线性码的最小距离和权重分布。
英文摘要：
We consider the simplified Algebraic Normal Form (sANF) of Boolean functions vanishing on Hamming spheres centred at zero and the associated sANF vector. We show that this vector is periodic, leading to an efficient computation of the sANF and to specific formulas for particular cases. Moreover, we explicitly provide a connection to the binary M{ö}bius transform of the elementary symmetric functions. We conclude by presenting a method based on polynomial evaluation to bound the minimum distance of binary nonlinear codes. The same method can be used to compute the minimum distance and the weight distribution of binary linear codes.
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