Designs from matroids

This paper presents new connections between designs and matroids. We generalize the Assmus–Mattson theorem and another coding-theoretical theorem with respect to matroids, and thereby present new sufficient conditions for obtaining $t$-designs from matroids. These conditions may be relaxed for some self-dual matroids. We use our results to prove new constructions of $t$-designs from linear codes, including several new extensions and variants of the Assmus–Mattson theorem. We also present weighted $t$-designs which generalize $t$-designs. New $t$-designs are obtained from our results.