Infinite families of 3-designs from special symmetric polynomials

Tang and Ding (IEEE Trans Inf Theory 67(1):244–254, 2021) opened a new direction of searching for t-designs from elementary symmetric polynomials, which are used to construct the first infinite family of linear codes supporting 4-designs. In this paper, we first study the properties of elementary symmetric polynomials with 6 or 7 variables over \(\textrm{GF}(3^{m})\) $\text{GF}(3^m)$ . Based on them, we present more infinite families of 3-designs that contain some 3-designs with new parameters as checked by Magma for small numbers m. We also construct an infinite family of cyclic codes over \(\textrm{GF}(q^2)\) $\text{GF}(q^2)$ and prove that the codewords of any nonzero weight support a 3-design. In particular, we present an infinite family of 6-dimensional AMDS codes over \(\textrm{GF}(3^{2m})\) $\text{GF}(3^{2m})$ holding an infinite family of 3-designs and an infinite family of 7-dimensional NMDS codes over \(\textrm{GF}(3^{2m})\) $\text{GF}(3^{2m})$ holding an infinite family of 3-designs.