  In this paper, the 3-rank of the incidence matrices of 2-designs supported by
the minimum weight codewords in a family of ternary linear codes considered in
[C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear
codes, Discrete Mathematics 340(10) (2017) 2415--2431] are computed. A lower
bound on the minimum distance of the ternary codes spanned by the incidence
matrices of these designs is derived, and it is proved that the codes are
subcodes of the 4th order generalized Reed-Muller codes.
