关于标准Reed-Solomon码的深洞猜想的注记

Reed-Solomon码是目前广泛应用于数字通信中的一类重要的极大距离可分码.Reed-Solomon码的译码过程通常采用最大似然译码算法.对于收到的一个码字u∈Fnq,最大似然译码算法关键在于确定码字u对于码C的错误距离d(u,C).熟知d(u,C)n-k,其中n,k分别为码C的码长和维数.若d(u,C)=n-k,则称u为码C的深洞.对于标准Reed-Solomon码,2012年洪和吴提出了一个著名的Wu-Hong深洞猜想.本文借助有限域Fq上极大距离可分码的生成矩阵,在一定条件下证明了标准Reed-Solomon码的Wu-Hong深洞猜想.

A remark on conjecture of deep holes of Standard Reed-Solomon codes

Reed-Solomon codes are now widely used in digital communication, which are an important class of maximum distance separable codes. we usually use the maximum likelihood decoding algorithm in the decoding process of Reed-Solomon codes. For the received word $u\in\mathbb{F}_q$, maximum likelihood decoding algorithm lies in determining its error distance $d(u,C)$. We have known that $d (u, C)\leq n-k$, where $n,k$ are the length and dimension of code $C$. If $d (u, C) =n-k$, then $u$ is called a deep hole of $C$. In 2012, Hong and Wu had proposed a famous deep hole conjecture of standard Reed-Solomon code. In this paper, we proved Wu-Hong conjecture of standard Reed-Solomon codes by using the generator matrix of maximum distance separable code.