  Construction of a large class of Mutually Unbiased Bases (MUBs) for non-prime
power composite dimensions ($d = k\times s$) is a long standing open problem,
which leads to different construction methods for the class Approximate MUBs
(AMUBs) by relaxing the criterion that the absolute value of the dot product
between two vectors chosen from different bases should be $\leq
\frac{\beta}{\sqrt{d}}$. In this chapter, we consider a more general class of
AMUBs (ARMUBs, considering the real ones too), compared to our earlier work in
[Cryptography and Communications, 14(3): 527--549, 2022]. We note that the
quality of AMUBs (ARMUBs) constructed using RBD$(X,A)$ with $|X|= d$,
critically depends on the parameters, $|s-k|$, $\mu$ (maximum number of
elements common between any pair of blocks), and the set of block sizes. We
present the construction of $\mathcal{O}(\sqrt{d})$ many $\beta$-AMUBs for
composite $d$ when $|s-k|< \sqrt{d}$, using RBDs having block sizes
approximately $\sqrt{d}$, such that $|\braket{\psi^l_i|\psi^m_j}| \leq
\frac{\beta}{\sqrt{d}}$ where $\beta = 1 + \frac{|s-k|}{2\sqrt{d}}+
\mathcal{O}(d^{-1}) \leq 2$. Moreover, if real Hadamard matrix of order $k$ or
$s$ exists, then one can construct at least $N(k)+1$ (or $N(s)+1$) many
$\beta$-ARMUBs for dimension $d$, with $\beta \leq 2 - \frac{|s-k|}{2\sqrt{d}}+
\mathcal{O}(d^{-1})< 2$, where $N(w)$ is the number of MOLS$(w)$. This improves
and generalizes some of our previous results for ARMUBs from two points, viz.,
the real cases are now extended to complex ones too. The earlier efforts use
some existing RBDs, whereas here we consider new instances of RBDs that provide
better results. Similar to the earlier cases, the AMUBs (ARMUBs) constructed
using RBDs are in general very sparse, where the sparsity $(\epsilon)$ is $1 -
\mathcal{O}(d^{-\frac{1}{2}})$.
