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Net replacements in the Hughes-Kleinfeld semifield planes. (English) Zbl 1201.51010

If \(\alpha\) is an automorphism of a field \(K\), the cone \(C_\alpha\) in \(PG(3,k)\) consists of the points \(\{ (x_0,x_1,x_2,x_3) \) \( \;| \;x_0^\alpha x_1 = x_2^{\alpha + 1}\}\) with vertex \(v_0 = (0,0,0,1).\) A set of planes of \(PG(3,k)\) which partitions these points without \(v_0\) is a flock of \(C_\alpha\). A flock is bilinear if the set of planes of a flock share precisely two lines. Recently, Cherowitzo found a class of bilinear flocks of the cone \(C_q\) in \(PG(3,q^2)\). The authors generalize these flocks to a large variety of bilinear flocks of the code \(C_q\), showing how they can be algebraically lifted from André spreads in \(PG(3,q).\) This gives a net replacement procedure in Hughes-Kleinfeld semifield planes of order \(q^4\) that constructs the set of André spreads.

MSC:

51E20 Combinatorial structures in finite projective spaces
51A40 Translation planes and spreads in linear incidence geometry
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References:

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