Hermitian unital

This example is partly derived from J. D. Key's paper ``Some applications of Magma in designs and codes: oval designs, hermitian unitals and generalized Reed-Muller codes'' (1995 unpublished manuscript). A unital (or unitary) design is a Steiner 2-design with parameters 2-. In the context of the desarguesian plane of square order , the set of absolute points and non-absolute lines of a unitary polarity forms a unital known as the hermitian unital.

Given a prime p and an integer m, the function below returns two values: the hermitian unital in PG, where ; and the design whose blocks are those intersections of lines of the plane with the unital which have cardinality q+1.

> HUnital := function(p, m)
>     q := p^m;
>     P, V, L := ProjectivePlane(q ^ 2);
>     hu := { pt : pt in V |
>         IsZero(pt[1]^(q+1) + pt[2]^(q+1) + pt[3]^(q+1)) };
>     blks := [blk : lin in L | #blk eq (q+1) where blk is lin meet hu];
>     return hu, Design< 2, SetToIndexedSet(hu) | blks >;
> end function;

We now evaluate the function for the case p=2, m=2, and compute the automorphism group for the resulting design:

> herm, D := HUnital(2, 2);
> IsUnital(herm);
true
> D;
2-(65, 5, 1) Design with 208 blocks
> IsSteiner(D, 2); 
true
> A := AutomorphismGroup(D);
> Order(A);
249600
> CompositionFactors(A);
    G
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  2A(2, 4)               = U(3, 4)
    1

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