论文标题
$ [q+1，2，q] $ mds代码的子场代码
The Subfield Codes of $[q+1, 2, q]$ MDS Codes
论文作者
论文摘要
最近，研究了$ 3 $ $ 3 $和$ 4 $的大型有限字段上的几何代码的子场代码，并获得了$ \ gf（p）$的距离 - 最佳子场代码，其中$ q = p^m $。在小字段上获得非常好的子字段代码的关键思想是，在具有小尺寸的扩展字段上选择非常好的线性代码。本文首先提出了$ [q+1，2，q] $ mds代码的一般结构，超过$ \ gf（q）$，然后在$ [q+1，2，q] $ mds $ \ gf（q）$的$ [q+1，2，q] $ mds中的某些$ \ gf（p）$上研究。获得了$ \ gf（p）$的两个尺寸 - 最佳代码的家族，并生产了几乎最佳代码的几个家庭。本文还提出了几个开放问题。
Recently, subfield codes of geometric codes over large finite fields $\gf(q)$ with dimension $3$ and $4$ were studied and distance-optimal subfield codes over $\gf(p)$ were obtained, where $q=p^m$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over $\gf(q)$, and then studies the subfield codes over $\gf(p)$ of some of the $[q+1, 2,q]$ MDS codes over $\gf(q)$. Two families of dimension-optimal codes over $\gf(p)$ are obtained, and several families of nearly optimal codes over $\gf(p)$ are produced. Several open problems are also proposed in this paper.
