SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2300-2317, January 2020.
Let $C$ denote a binary linear code with length $n$ all of whose coordinates are essential; i.e., for each coordinate there is a codeword that is not zero in that position. Then the maximum distance $D$ is strictly bigger than $n/2$, and the extremum $D=(n+1)/2$ is attained exactly by punctured Hadamard codes. In this paper, we classify binary linear codes with $D=n/2+1$. All of these codes can be produced from punctured Hadamard codes in one of essentially three different ways, each having a transparent description.

中文翻译：

---

具有接近极值最大距离的二进制线性代码

SIAM离散数学杂志，第34卷，第4期，第2300-2317页，2020年1月。
令$ C $表示长度为$ n $的二进制线性代码，所有坐标都是必不可少的。即，对于每个坐标，在该位置存在一个不为零的代码字。然后，最大距离$ D $严格大于$ n / 2 $，并且极值$ D =（n + 1）/ 2 $是通过打孔的Hadamard码精确获得的。在本文中，我们用$ D = n / 2 + 1 $对二进制线性代码进行分类。所有这些代码都可以从打孔的Hadamard代码以基本上三种不同的方式之一生成，每种方式都具有透明的描述。