A characterization of the primals in PG(m, 2)
Designs, Codes and Cryptography
The lines of PG(4, 2) are the points on a quintic in PG(9, 2)
Journal of Combinatorial Theory Series A
Designs, Codes and Cryptography
The ψ-associate X# of a flat X in PG(n, 2)
Designs, Codes and Cryptography
The cubic Segre variety in PG(5, 2)
Designs, Codes and Cryptography
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The 155 points of the Grassmannian G1, 4, 2 of lines of PG (4, 2) = PV(5, 2) are those points x ∈ PG (9, 2) = P ( ∧2V(5, 2)) which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X ⊆ PG (9, 2) will be termed odd or even according as X intersects C1, 4, 2 in an odd or even number of points. Let Q‡ (x1, …, x5) denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X ⊆ PG (9, 2) by \qquad Y = {y ∈ PG (n, 2) | Q‡ (x1, x2, x3, x4, y) = 0, for all x1, x2, x3, x4 ∈ X }. Because Q‡ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X ⊆ X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that (X#)# = X is provided by any 4-flat X which is external to G1, 4, 2}. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and (X#)# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which X### = X.