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Symplectic instanton vector bundles on the projective space $\mathbb{P}^3$constitute a natural generalization of mathematical instantons of rank 2. Westudy the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vectorbundles on $\mathbb{P}^3$ with $r\ge2$ and second Chern class $n\ge r,\ n\equivr({\rm mod}2)$. We give an explicit construction of an irreducible component$I^*_{n,r}$ of this space for each such value of $n$ and show that $I^*_{n,r}$has the expected dimension $4n(r+1)-r(2r+1)$.

Moduli of symplectic instanton vector bundles of higher rank on projective space $\mathbb{P}^3$