标题： 一个无限的线性码族支持4设计 显示英文标题

标题： An infinite family of linear codes supporting 4-designs

摘要： 第一个支持$4$设计的线性码是$[11, 6, 5]$三元 Golay 码，该码于1949年由Golay发现。 在过去71年中，发现了许多零散的线性码，它们包含$4$设计或$5$设计，同时构造了许多支持$3$设计的线性码的无限族。 然而，关于是否存在一个线性码的无限族，其包含一个无限族的$t$设计，对于$t\geq 4$的问题在过去的71年里仍然悬而未决。 本文通过提出长度为$2^{2m+1}+1$的$\mathrm{GF}(2^{2m+1})$上的 BCH 码的无限族，解决了这个长期存在的问题，该码持有$4$-$(2^{2m+1}+1, 6, 2^{2m}-4)$设计的无限族。 此外，还提出了一个包含球面设计$S(3, 5, 4^m+1)$的线性码的无限族。 显示英文摘要

摘要： The first linear code supporting a $4$-design was the $[11, 6, 5]$ ternary Golay code discovered in 1949 by Golay. In the past 71 years, sporadic linear codes holding $4$-designs or $5$-designs were discovered and many infinite families of linear codes supporting $3$-designs were constructed. However, the question as to whether there is an infinite family of linear codes holding an infinite family of $t$-designs for $t\geq 4$ 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. Moreover, an infinite family of linear codes holding the spherical design $S(3, 5, 4^m+1)$ is presented.