The question as to whether there exists an infinite family of near MDS codes holding an infinite family of $t$ -designs for $t\geq 2$ was answered in the recent paper [Infinite families of near MDS codes holding $t$ -designs, IEEE Trans. Inf. Theory 66(9) (2020)], where an infinite family of near MDS codes holding an infinite family of 3-designs and an infinite family of near MDS codes holding an infinite family of 2-designs were presented, but no infinite family of linear codes holding an infinite family of 4-designs was presented. Hence, the question as to whether there is an infinite family of linear codes holding an infinite family of 4-designs remains open for 71 years. This paper settles this long-standing problem by presenting an infinite family of BCH codes of length $2^{2m+1}+1$ over ${\mathrm {GF}}(2^{2m+1})$ holding an infinite family of 4- $(2^{2m+1}+1, 6, 2^{2m}-4)$ designs. This paper also provides another solution to the first question, as some of the BCH codes presented in this paper are also near MDS. Moreover, an infinite family of linear codes holding the spherical geometry design $S(3, 5, 4^{m}+1)$ is presented. The new direction of searching for $t$ -designs with elementary symmetric polynomials will be further advanced.

中文翻译：

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支持 4 种设计的无限线性代码系列

关于是否存在持有无限族的近 MDS 码的无限族的问题 $t$ -设计为 $t\geq 2$ 在最近的论文中得到了回答 [Infinite family of near MDS code holding $t$ -设计，IEEE Trans。信息 理论 66(9) (2020)]，其中提出了无限系列的近 MDS 代码持有无限系列的 3-设计和无限系列的近 MDS 代码持有无限系列的 2-设计，但没有无限系列提出了包含无限系列 4 设计的线性代码。因此，是否存在包含无限 4 设计系列的无限线性代码系列的问题仍然存在 71 年。本文通过提出无限长的 BCH 码族来解决这个长期存在的问题 $2^{2m+1}+1$ 超过 ${\mathrm {GF}}(2^{2m+1})$ 抱着无限的四口之家 $(2^{2m+1}+1, 6, 2^{2m}-4)$ 设计。本文还为第一个问题提供了另一种解决方案，因为本文中提出的一些 BCH 代码也接近 MDS。此外，拥有球面几何设计的无限线性代码族 $S(3, 5, 4^{m}+1)$ 被表达。寻找新方向 $t$ - 具有基本对称多项式的设计将得到进一步推进。