  This paper is concerned with the affine-invariant ternary codes which are
defined by Hermitian functions. We compute the incidence matrices of 2-designs
that are supported by the minimum weight codewords of these ternary codes. The
linear codes generated by the rows of these incidence matrix are subcodes of
the extended codes of the 4-th order generalized Reed-Muller codes and they
also hold 2-designs. Finally, we give the dimensions and lower bound of the
minimum weights of these linear codes.
