Coding theory and $t$-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these three-weight subcodes. These shortened and punctured codes contain some codes that have the same parameters as the best ternary linear codes known in the database maintained by Markus Grassl at http://www.codetables.de/. These three-weight subcodes with a special weight distribution do not satisfy the conditions of the Assmus-Mattson theorem and do not admit $2$-transitive or $2$-homogeneous automorphism groups in general. By employing the theory of projective geometries and projective generalized Reed-Muller codes, we prove that they still hold $2$-designs. We also determine the parameters of these $2$-designs. This paper mainly confirms some recent conjectures of Ding and Li regarding Steiner systems and $2$-designs from a special type of ternary projective codes.