论文标题
一个支持4设计的无限线性代码家族
An infinite family of linear codes supporting 4-designs
论文作者
论文摘要
支撑$ 4 $ -Design的第一个线性代码是Golay在1949年发现的$ [11，6，5] $ Ternary Golay代码。在过去的71年中，发现了零星的零星线性代码，持有$ 4 $ -Designs或$ 5 $ - 设计，并且构建了许多支持$ 3 $ designs的无限线性代码系列。但是，关于是否有一个无限的线性代码家族的疑问，持有一个无限的$ t $ dessigns $ t \ geq 4 $的家族，持续了71年。本文通过介绍一个长度为$ 2^{2m+1}+1 $ $ \ MATHRM {gf}（2^{2m+1}）的无限型BCH代码的家族来解决这个长期存在的问题。此外，提出了一个无限的线性代码家族，其中包含球形设计$ s（3，5，4^m+1）$。
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.
