Infinite families of 2-designs and 3-designs from linear codes

The interplay between coding theory and t-designs started many years ago. While every t-design yields a linear code over every finite field, the largest t for which an infinite family of t-designs is derived directly from a linear or nonlinear code is t=3. Sporadic 4-designs and 5-designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of 2-designs and 3-designs from linear codes. The parameters of some known t-designs are also derived. In addition, many conjectured infinite families of 2-designs are also presented.