I'm working on the Shortest Lattice Vector Problem (SVP) for a paper that I'm currently writing. I wish to verify whether a particular structural, namely the **building block property** ( refer to the paper by Prof. Manindra Agrawal titled *"Universal Relation"*), is satisfied by the SVP.

Basically what this needs me to do is the following:

- Come up with a lattice basis $\mathcal{B} \in \mathcal{Z}^{n\times n}$, which is of
*full rank*and of dimension: $n > 3$ - From the coefficient matrix $\mathcal{X} \in \mathcal{Z}^{n}$, of the vector $\mathcal{BX}$ in the lattice, choose any $3$ coefficients, which remain the same for all $\mathcal{BX}$.
- These $3$ coefficients can take a total of $8$ possible values, i.e. if the coefficients $(x_i,x_j,x_k)= (1,0,1)$ implies that $x_i$ and $x_k$ take a non-zero value and $x_j$ is zero.
- Now for all of the 8 possible values there should
**exist atleast one solution of the SVP**(that is there should be a shortest lattice vector whose coefficients satisfy the required constraints)**except for the case**where all the three coefficients take the value 0.

The problem that I'm facing is that I'm finding it hard to come up with such a construction.

- Could you suggest a way I could proceed in constructing such a lattice?
- Or is it even possible to construct such a lattice basis, a possible heuristic argument would also help?
- Would it be possible if we change the fact that $\mathcal{B} \in \mathcal{R}^{n \times n}$? If yes then how would I proceed?

PS: I had initially tried an approach (Minimum number of points required to determine lattice basis), to move in the reverse direction, that I first decided upon 7 shortest vectors and then tried to deduce the a possible satisfying lattice basis. But based on the comments on the question it seems that it is a dead end.