Base of $(n+1)$ elements in a lattice

Does there exist a lattice in $\mathbb{R}^n$, with an independent generative family $(b_1, \dots, b_{n+1})$ of $(n+1)$ vectors (without any loss of generality I suppose $(b_1, \dots, b_{n})$ is a $\mathbb{R}$-basis), for no generative family of size $n$.

I know:

• If these vectors are in $\mathbb{Q}^n$, then the answer is NO, because the lattice is included in $\frac{1}{q}\mathbb{Z}^n$ with $q\in \mathbb{N},$ and $\frac{1}{q}\mathbb{Z}^n$ is a free $\mathbb{Z}-$module of rank $n$, (and a sub module of a free module is free with rank lower than the bigger).

• It exists $(\lambda_1, \lambda_2, \dots, \lambda_n, \lambda_{n+1})\in \mathbb{Z}^{n+1}$, such that $\sum^{n+1}_{i=1} \lambda_i b_i=0.$ (Else, we can create an infinite number of points in the fundamental parallepiped of $(b_1, \dots, b_n).)$

But I do neither know how deduce the existence of a generative family of size $n$, nor to show the existence of base (because a potential family of size $(n+1)$ is clearly not).

Two things

• One can always find a generating set of a lattice of size at most the dimension of the ambient space (so you can always find an $n$-dimensional generating set for a lattice in $\mathbb{R}^n$)
• One can always reduce a generating set of a lattice to a basis (there are standard algorithms, namely for computing what is called the Hermite Normal Form).

Your question does have an interesting component to it though. In particular, many lattice problems (such as the Shortest Independent Vectors Problem) are phrased in terms of sets of independent vectors (that generate a subspace $E$ of a certain rank, but need not be a basis for $L\cap E$). For a particular example, there is an explicit 10-dimensional lattice known such that

• that lattice is generated by its minimal vectors, but
• it (provably) has no basis of minimal vectors.

It is known that 10 is the lowest dimension this can occur in. Note that one can bound the gap between the (product of the) norms of elements in a minimal generating set of a lattice, see Hermite vs Minkowski by Martinet.

This has some relevance to cryptography, as certain lattice algorithms (either Babai's nearest planes or Babai rounding, I forget) do not require a basis to function, but only a generating set. I've seen some authors (I believe one of Chris Peikert's papers) use this insight for a particular lattice, I believe $D_n^* = 2\mathbb{Z}^n + (1,1,\dots,1)^t\cdot\mathbb{Z}$, but I would have to check. Specifically, by instantiating the algorithm with a short set of independent vectors (vs a short basis), one can sometimes get better algorithmic performance.

You just need to use LLL algorithm for $(b_1, \dots, b_{n+1})$. More specifically, In BGPS21,the Algorithm 1 and Theorem 3.6 gave a more efficient algorithm than LLL.