Generators for Vector Spaces Spanned by Double Zeta Values with Even Weight

Let ${\mathcal{D}\mathcal{Z}}_k$ be the $\mathbb{Q}$-vector space spanned by double zeta values with weight $k$, and ${\mathcal{D}\mathcal{M}}_k$ be its quotient space divided by the space ${\mathcal{P}\mathcal{Z}}_k$ spanned by the zeta value $\zeta (k)$ and products of two zeta values with total weight $k$. When $k$ is even, an upper bound for the dimension of ${\mathcal{D}\mathcal{M}}_k$ is known. By adding the dimensions of ${\mathcal{D}\mathcal{M}}_k$ and ${\mathcal{P}\mathcal{Z}}_k$, an upper bound of ${\mathcal{D}\mathcal{Z}}_k$ which equals $k\mathrm{/}2$ minus the dimension of the space of modular forms of weight $k$ on the modular group is given. In this note, we obtain some specific sets of generators for ${\mathcal{D}\mathcal{M}}_k$ which represent the upper bound. These yield the corresponding sets and the upper bound for ${\mathcal{D}\mathcal{Z}}_k$.