一类支持4-设计的线性码
An infinite family of linear codes supporting 4-designs
摘要：
第一个支持$4$设计的线性码是Golay在1949年发现的$[11, 6, 5]$三进制Golay码.在过去的71年中，人们发现了具有$4$ -设计或$5$ -设计的零星线性码，并构造了许多支持$3$ -设计的线性码的无限族.然而，关于是否有一个无限族的线性代码持有一个无限族的$t$ -设计$t\geq 4$仍然开放了71年的问题。本文提出了一个无限族的BCH码的长度$2^{2m+1}+1$在$\mathrm{GF}(2^{2m+1})$持有一个无限族的$4$ - $(2^{2m+1}+1, 6, 2^{2m}-4)$设计，解决了这个长期存在的问题。此外，提出了一个无限族的线性码保持球形设计$S(3, 5, 4^m+1)$。
英文摘要：
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.
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