A construction based on a 4l×4l\documentclass[12pt]{minimal}
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\begin{document}$$4l \times 4l$$\end{document} Hadamard matrix leads to a new family of optimal orthoplex packings in Grassmannian spaces GR(8l,4l)\documentclass[12pt]{minimal}
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\begin{document}$$G_{\mathbb {R}}(8l, 4l)$$\end{document} and GC(4l,2l)\documentclass[12pt]{minimal}
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\begin{document}$$G_{\mathbb {C}}(4l, 2l)$$\end{document}. A related construction gives an optimal simplex packings in GR(8l-1,4l-1)\documentclass[12pt]{minimal}
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\begin{document}$$G_{\mathbb {R}}(8 l-1, 4 l - 1)$$\end{document} and GR(8l-1,4l)\documentclass[12pt]{minimal}
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\begin{document}$$G_{\mathbb {R}}(8l-1, 4l)$$\end{document} with the additional assumption of an 8l×8l\documentclass[12pt]{minimal}
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\begin{document}$$8l \times 8l$$\end{document} skew Hadamard matrix and a related 1-factorization of a complete graph. A construction of a maximal optimal simplex packings in GC(2l-1,l-1)\documentclass[12pt]{minimal}
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\begin{document}$$G_{\mathbb {C}}(2l-1, l- 1)$$\end{document} and GC(2l-1,l)\documentclass[12pt]{minimal}
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\begin{document}$$G_{\mathbb {C}}(2l-1,l)$$\end{document} is given.
