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Coding theory and $t$-designs have close connections and interestinginterplay. In this paper, we first introduce a class of ternary linear codesand study their parameters. We then focus on their three-weight subcodes with aspecial weight distribution. We determine the weight distributions of someshortened codes and punctured codes of these three-weight subcodes. Theseshortened and punctured codes contain some codes that have the same parametersas the best ternary linear codes known in the database maintained by MarkusGrassl at http://www.codetables.de/. These three-weight subcodes with a specialweight distribution do not satisfy the conditions of the Assmus-Mattson theoremand do not admit $2$-transitive or $2$-homogeneous automorphism groups ingeneral. By employing the theory of projective geometries and projectivegeneralized Reed-Muller codes, we prove that they still hold $2$-designs. Wealso determine the parameters of these $2$-designs. This paper mainly confirmssome recent conjectures of Ding and Li regarding Steiner systems and$2$-designs from a special type of ternary projective codes.

Steiner systems $S(2, 4, \frac{3^m-1}{2})$ and $2$-designs from ternary linear codes of length $\frac{3^m-1}{2}$