I have some questions on lower bounds on the rank of unimodular lattices given the bilinear pairing of a subset of its basis is known.

Let $\Lambda$ be an odd, unimodular matrix of signature $(1,T)$. In particular, there exists a basis in which the bilinear form $B$ of the lattice is given by \begin{equation} B = \rm diag(1,-1,\cdots,-1) \,. \end{equation} Now let us assume that there exists $k$ mutually orthogonal lattice vectors $x_1,\cdots,x_k$ with norm $-n < -1$, i.e., \begin{equation} x_i \cdot x_j = -n \delta_{ij} \,, \end{equation} that are further assumed to be elements of a basis of the lattice $\Lambda$. Given the existence of these vectors, how big does $T$ have to be to accommodate them? In other words, what is the lower bound $b(n,k)$ on $T$?

There are some trivial bounds we can get. For example, \begin{equation} b(n,k) \geq k \,. \end{equation} Also, it is clear that \begin{equation} b(n,k) \leq nk \,. \end{equation} Can we get a tighter range or better lower-bounds on $b(n,k)$? For example, it can be shown that \begin{equation} b(n,k) \geq 2k \,. \end{equation} Can one do better? (One might mistakenly think that $b(m^2,k)=k < 2k$, but the assumption is that $\{ x_i : i =1,\cdots, k \}$ is a subset of the basis of the lattice. Hence, for example, $b(4,1) = 3$, rather than $1$.)

Can one find better linear lower-bounds? In other words, does there exist a number $\alpha_n >2$ such that \begin{equation} b(n,k) \geq \alpha_n k +\beta \end{equation} for some constant $\beta$?

Now let us assume there exist $k$ mutually orthogonal basis vectors $x_1, \cdots, x_k$ of $\Lambda$, all of which have negative norms, i.e., \begin{equation} (x_i \cdot x_j) = {\rm diag} (-n_1, \cdots , -n_k) \,, \end{equation} with $n_i > 1$. What is the smallest allowed value $b(n_1,\cdots,n_k)$ for $T$ in this case?

Can one imagine there being good linear bound(s) \begin{equation} b(n_1,\cdots,n_k) \geq \gamma_{n_1} + \cdots + \gamma_{n_k} +\beta \end{equation} for some $\beta$ in this more generalized case as well? In other words, does there exist $\gamma_n >1$ for each $n$ that gives such linear bounds?

I have only been able to get some very loose bounds using only very elementary tools so far. Where should I look to learn methods that can be used to effectively tackle with problems of this sort?

Thank you!