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 \(\textrm{GF}(3^{m})\). 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 \(\textrm{GF}(q^2)\) 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 \(\textrm{GF}(3^{2m})\) holding an infinite family of 3-designs and an infinite family of 7-dimensional NMDS codes over \(\textrm{GF}(3^{2m})\) holding an infinite family of 3-designs.

Tang 和 Ding（IEEE Trans Inf Theory 67(1):244–254, 2021）开辟了从初等对称多项式中搜索t设计的新方向，该多项式用于构造第一个支持 4 设计的无限线性码族。在本文中，我们首先研究\(\textrm{GF}(3^{m})\)上具有 6 或 7 个变量的初等对称多项式的性质。基于它们，我们提出了更多无限的 3 设计系列，其中包含一些具有新参数的 3 设计，由 Magma 检查小数字m 。我们还在\(\textrm{GF}(q^2)\)上构造了无限族循环码，并证明任何非零权重的码字都支持 3 设计。特别是，我们在\(\textrm{GF}(3^{2m})\)上提出了无限系列的 6 维 AMDS 代码，其中包含无限系列的 3 设计和无限系列的 7 维 NMDS 代码\(\textrm{GF}(3^{2m})\)拥有无限的 3 设计系列。