A linear code with parameters of the form $[n,k,n-k+1]$ is referred to as an MDS (maximum distance separable) code. A linear code with parameters of the form $[n,k,n-k]$ is said to be almost MDS (i.e., almost maximum distance separable) or AMDS for short. A code is said to be near maximum distance separable (in short, near MDS or NMDS) if both the code and its dual are almost maximum distance separable. Near MDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. There are many unsolved problems about near MDS codes. It is hard to construct an infinite family of near MDS codes whose weight distributions can be settled. In this paper, seven infinite families of $[2^m+1,3,2^m-2]$ near MDS codes over $\mathrm{GF}\left(2^m\right)$ and seven infinite families of $[2^m+2,3,2^m-1]$ near MDS codes over $\mathrm{GF}\left(2^m\right)$ are constructed with special oval polynomials for odd $m$. In addition, nine infinite families of optimal $[2^m+3,3,2^m]$ near MDS codes over $\mathrm{GF}\left(2^m\right)$ are constructed with oval polynomials in general. The weight distributions of these near MDS codes are settled.

参数形式为线性的代码 $[ñ，ķ，ñ-ķ+\mathrm{1个}]$被称为MDS（最大距离可分离）代码。参数形式为线性的代码$[ñ，ķ，ñ-ķ]$据说几乎是MDS（即几乎可分离的最大距离）或简称AMDS。如果代码及其对偶几乎都是最大可分离距离，则称该代码接近最大可分离距离（简而言之，接近MDS或NMDS）。接近MDS的代码对应于有限几何中有趣的对象，并在组合和密码学中有很好的应用。关于近MDS代码有许多未解决的问题。很难构造可以确定权重分布的无限近MDS代码族。本文中的七个无穷家族$[2^{米}+\mathrm{1个}，3，2^{米}-2]$ 接近MDS代码 $\mathrm{GF}（2^{米}）$ 和七个无限的家庭 $[2^{米}+2，3，2^{米}-\mathrm{1个}]$ 接近MDS代码 $\mathrm{GF}（2^{米}）$ 用特殊的椭圆多项式构造奇数 $米$。此外，还有9个无限的最优族$[2^{米}+3，3，2^{米}]$ 接近MDS代码 $\mathrm{GF}（2^{米}）$通常用椭圆多项式构造。确定这些接近MDS代码的权重分布。