Based on maximally entangled states, we explore the constructions of mutually
unbiased bases in bipartite quantum systems. We present a new way to construct
mutually unbiased bases by difference matrices in the theory of combinatorial
designs. In particular, we establish $q$ mutually unbiased bases with $q-1$
maximally entangled bases and one product basis in $\mathbb{C}^q\otimes
\mathbb{C}^q$ for arbitrary prime power $q$. In addition, we construct
maximally entangled bases for dimension of composite numbers of non-prime
power, such as five maximally entangled bases in $\mathbb{C}^{12}\otimes
\mathbb{C}^{12}$ and $\mathbb{C}^{21}\otimes\mathbb{C}^{21}$, which improve the
known lower bounds for $d=3m$, with $(3,m)=1$ in $\mathbb{C}^{d}\otimes
\mathbb{C}^{d}$. Furthermore, we construct $p+1$ mutually unbiased bases with
$p$ maximally entangled bases and one product basis in $\mathbb{C}^p\otimes
\mathbb{C}^{p^2}$ for arbitrary prime number $p$.
