I'm looking for a way to generate randomly a set of $N$ hyperplanes equations using the following equation: $,\sum\limits^{k-1}_{i=0} (a_i\times x_i) = b\text{, } k\in \mathbb{N/\{0,1\}}$. And $a_i,x_i \in \mathbb{N}$.
In a way, for any arrangements of $k$ equations of this set, I've always a solution to the linear system.
I've already tried to generate them by defining some rules:
- uniqueness of equations;
- autorize 1 zero in each column;
- forbid the equation with only zero;
- $\forall i \in \{0,k-1\}, \text{ } a_i = 0$).
- forbid the equation as multiple of other equations:
- $\forall i \in \{0,k-1\},\text{ }\exists j \in \{0,k-1\}/\{i\}, \text{ }L_i = \delta \times l_j (\text{where }\delta \in \mathbb{N}$).
The algorithm is very gross then I've noticed I needed to also exclude equations as: $L_h = \delta\times L_i + \sigma \times L_j$. So I've started to model it as a CSP.
If you have any ideas to generate this set of solvable hyperplanes equations or new rules for my CSP I'm very interested !
Thanks you in advance.