Abstract
We construct a homogeneous full projective embedding of the dual polar space \(\textit{DW}(2n-1,2)\) from the hyperplane intersections of hyperbolic type of the parabolic quadric Q(2n, 2). We believe that this embedding is universal, but have not succeeded in proving that. As a by-product of our investigations, we have obtained necessary and sufficient conditions for this to be the case and came across two other homogeneous full projective embeddings of \(\textit{DW}(2n-1,2)\), one with vector dimension \(\frac{2^{2n-1} + 3 \cdot 2^{n-1} -2}{3}\) and another one with vector dimension \(\frac{2^{2n-1} + 3 \cdot 2^{n-1} -2 - 6n}{3}\).
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Acknowledgements
The author wishes to thank Peter Vandendriessche for performing the computer computations which showed that the hyperbolic embedding \(\epsilon _h\) of \(\textit{DW}(2n-1,2)\) is universal for \(n \le 7\). He also wishes to thank Bert Seghers for discussions on the topic of this paper.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.
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De Bruyn, B. Three homogeneous embeddings of \(\textit{DW}(2n-1,2)\). Des. Codes Cryptogr. 87, 909–929 (2019). https://doi.org/10.1007/s10623-018-0531-0
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DOI: https://doi.org/10.1007/s10623-018-0531-0