Example Plane_unital (H154E11)

The following code computes the Hermitian unital given by the equation x^{q + 1} + y^{q + 1} + z^{q + 1} = 0 in PG₂(q²) for q = 3.

> q := 3;
> F<w> := GaloisField(q ^ 2);
> P, V, L := FiniteProjectivePlane(F);
> 
> hu := { V | [x,y,z] : x, y, z in F |                                     
>                x^(q+1) + y^(q+1) + z^(q+1) eq 0 and {x, y, z} ne {0} };
> 
> IsUnital(P, hu);
true
> UnitalFeet(P, hu, V.1);    
{  ( 0 : 1 : w ), ( 0 : 1 : w^3 ), ( 0 : 1 : w^5 ), ( 0 : 1 : w^7 ) }

Since this set has more than one element, V.1 must not be in hu:

> V.1 in hu;
false

For a point in hu:

> UnitalFeet(P, hu, Rep(hu));
{  ( 1 : 0 : w^7 ) }

Now we construct the design given by hu.

> blks := [blk : lin in L | #blk eq (q+1) where blk is lin meet hu ];
> D := Design< 2, SetToIndexedSet(hu) | blks >;
> D;
2-(28, 4, 1) Design with 63 blocks

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