论文标题
来自椭圆形多项式的MDS代码附近
Near MDS codes from oval polynomials
论文作者
论文摘要
具有$ [N，K，N-K+1] $的参数的线性代码称为MDS（最大距离可分开）代码。具有$ [n，k，n-k] $的参数的线性代码几乎是MDS（即几乎最大距离可分开）或简称AMDS。如果代码及其双重距离几乎可分开，则据说代码接近最大距离（简而言之接近MDS或NMD）。近MDS代码对应于有限几何形状中有趣的对象，并在组合学和加密中具有很好的应用。在本文中，$ [2^m+1、3、2^m-2] $的七个无限家庭接近$ \ gf（2^m）$和七个无限家庭的$ [2^m+2、3、3、2^m-1] $附近的MDS代码与$ \ gf（2^m）$相近的MDS代码与Special Oval Polynomials ofd Odd $ m $ MD构建。此外，在$ \ gf（2^m）$上的MDS代码附近的9个无限族的最佳$ [2^m+3，3，2^m] $通常用椭圆形多项式构建。
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. In this paper, seven infinite families of $[2^m+1, 3, 2^m-2]$ near MDS codes over $\gf(2^m)$ and seven infinite families of $[2^m+2, 3, 2^m-1]$ near MDS codes over $\gf(2^m)$ 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 $\gf(2^m)$ are constructed with oval polynomials in general.
