The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
The Polynomial Degree of the Grassmannian $${\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}$$
Designs, Codes and Cryptography
The ψ-associate X# of a flat X in PG(n, 2)
Designs, Codes and Cryptography
On invariant notions of Segre varieties in binary projective spaces
Designs, Codes and Cryptography
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We consider various aspects of the Segre variety $${\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}$$ in PG(7, 2), whose stabilizer group $${\mathcal{G}_{\mathcal{S}} has the structure $${\mathcal{N}\rtimes{\rm Sym}(3),}$$ where $${\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}$$ In particular we prove that $${\mathcal{S}}$$ determines a distinguished Z 3-subgroup $${\mathcal{Z} such that $${A\mathcal{Z}A^{-1}=\mathcal{Z},}$$ for all $${A\in\mathcal{G}_{\mathcal{S}},}$$ and in consequence $${\mathcal{S}}$$ determines a $${\mathcal{G}_{\mathcal{S}}}$$ -invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties $${\mathcal{S}_{1,1,1}(2)}$$ in PG(7, 2) come along in triplets $${\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}$$ which share the same distinguished Z 3-subgroup $${\mathcal{Z} We conclude by determining all fifteen $${\mathcal{G}_{\mathcal{S}}}$$ -invariant polynomial functions on PG(7, 2) which have degree $${\mathcal{G}_{\mathcal{S}}}$$ -orbits of points in PG(7, 2).