Steiner systems S(2, 4, 3m-1/2) and 2-designs from ternary linear codes of length 3m-1/2

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 . 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.