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In this paper, the 3-rank of the incidence matrices of 2-designs supported bythe 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 linearcodes, Discrete Mathematics 340(10) (2017) 2415--2431] are computed. A lowerbound on the minimum distance of the ternary codes spanned by the incidencematrices of these designs is derived, and it is proved that the codes aresubcodes of the 4th order generalized Reed-Muller codes.

Linear codes of 2-designs associated with subcodes of the ternary generalized Reed-Muller codes