# Constructing Disjoint Sets with Product Constraints in a Large Field" [closed]

Let $$n$$ be an integer. Let there be $$n$$ integer weights $$w_1,w_2,\ldots,w_n$$. Let there be a large field $$\mathbb{F}$$. We can assume that $$n\ll|\mathbb{F}|$$ and $$\sum_{i\in {1,2,\ldots,n}} w_i \ll |\mathbb{F}|$$ .

Are there algorithms that on input $$n$$ and the weights, outputs $$n$$ different sets $$S_1,S_2,...,S_n$$, and $$n^2$$ field elements $$k_{i,j}\ne 0$$ for $$i,j={1,2,\ldots,n}$$ that satisfy the following constraints.

1. $$S_i\subseteq \mathbb{F}$$ and $$|S_i|=w_i$$, i.e., $$S_i$$ consists of $$w_i$$ elements.
2. The sets are disjoint.
3. For all pair of sets $$(S_i,S_j)$$, for any given $$a\in S_i$$, $$\prod_{b\in S_j} (a-b)=k_{i,j}$$ for some non-zero $$k_{i,j}$$.

Any pointers to where to look for such an algorithm will be really helpful. Thank you!

• Are you given $k_{i,j}$ or simply requiring that it be non zero? Commented Jul 15, 2023 at 9:52
• $k_{i,j}$ can be arbitrary non-zero values output by the algorithm. I updated the question to include this information. Commented Jul 15, 2023 at 16:21