Designs with block-size 6 in projective planes of characteristic 2

We construct simple 3-designs and 4-designs of block-size 6 in the classical projective planesPG(2,q),q a power of 2. All of our designs are invariant under the projective groupPGL(3,q). Aside from several infinite series of 3-designs we get some relatively small designs of independent interest, e.g. designs with parameters 4-(21, 6, 16) and 4-(73, 6, 330) defined in the planes of orders 4 and 8, respectively.