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 $1 - \mathcal{O}(d^{-\frac{1}{2}})$.

翻译：对于非素数幂复合维度（$d = k \times s$）的大类互无偏基（MUBs）的构造长期以来是一个未解决的开放问题。通过放松不同基中选取的两个向量点积绝对值需满足 $\leq \frac{\beta}{\sqrt{d}}$ 的条件，可衍生出近似互无偏基（AMUBs）的不同构造方法。本章考虑比我们先前工作[Cryptography and Communications, 14(3): 527--549, 2022]更广义的AMUBs类别（ARMUBs，同时包含实数情形）。研究发现，基于RBD$(X,A)$（其中$|X|= d$）构造的AMUBs（ARMUBs）质量关键取决于参数$|s-k|$、$\mu$（任意两个块之间最大公共元素数量）以及块大小集合。当$|s-k|<\sqrt{d}$时，我们利用块大小近似为$\sqrt{d}$的RBDs，为复合维度$d$构造了$\mathcal{O}(\sqrt{d})$个$\beta$-AMUBs，使得$|\braket{\psi^l_i|\psi^m_j}| \leq \frac{\beta}{\sqrt{d}}$，其中$\beta = 1 + \frac{|s-k|}{2\sqrt{d}}+ \mathcal{O}(d^{-1}) \leq 2$。进一步地，若存在$k$阶或$s$阶实哈达玛矩阵，则可构造至少$N(k)+1$（或$N(s)+1$）个维度$d$的$\beta$-ARMUBs，满足$\beta \leq 2 - \frac{|s-k|}{2\sqrt{d}}+ \mathcal{O}(d^{-1})< 2$，这里$N(w)$表示MOLS$(w)$的数量。该结果从两方面改进并推广了先前关于ARMUBs的部分结论：实数情形现已扩展至复数情形。早期工作使用现有RBDs，而本文考虑能提供更优结果的新型RBDs实例。与前期案例类似，基于RBDs构造的AMUBs（ARMUBs）通常高度稀疏，稀疏度$(\epsilon)$为$1 - \mathcal{O}(d^{-\frac{1}{2}})$。