标准Reed-Solomon码的错误距离

Error distance of standard Reed-Solomon codes

摘要:

标准Reed-Solomon码的错误距离在其译码过程中发挥着重要的作用. 在2012年, Hong和Wu提出了一个著名的错误距离猜想. 本文借助有限域上的二次型理论, 通过计算极大距离可分码的生成矩阵, 推得奇特征有限域$\mathbb{F}_q$上一类$q-4$次多项式定义的码字不是标准Reed-Solomon码的深洞, 从而部分证明了标准Reed-Solomon 码的错误距离猜想.

Abstract:

Error distance plays an important role in the decoding of standard Reed-Solomon codes. In 2012, Hong and Wu proposed a famous conjecture of standard Reed-Solomon codes on error distance. In this paper, We show that some polynomials of degree $q-4$ can not define deep holes over finite fields with odd characteristic by using the quadratic form and the generator matrix of maximum distance separable codes. In fact, we partially proved error distance conjecture of standard Reed-Solomon codes.