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We study the unextendible maximally entangled bases (UMEB) in$\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and connect it with the partialHadamard matrix. Firstly, we show that for a given special UMEB in$\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, there is a partial Hadamard matrixcan not extend to a complete Hadamard matrix in $\mathbb{C}^{d}$. As acorollary, any $(d-1)\times d$ partial Hadamard matrix can extend to a completeHadamard matrix. Then we obtain that for any $d$ there is an UMEB except $d=p\\text{or}\ 2p$, where $p\equiv 3\mod 4$ and $p$ is a prime. Finally, we arguethat there exist different kinds of constructions of UMEB in$\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}$ for any $n\in \mathbb{N}$ and$d=3\times5 \times7$.

Connecting the UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ with partial Hadamard matrices