环F_2+uF_2+u^2F_2上线性码及其Gray象的生成矩阵

Generator matrix of the linear codes and Gray images over the ring F_2+uF_2+u^2F_2

文章定义了环F2+uF2+u2F2与F2之间的一种新Gray映射,利用环F2+uF2+u2F2上线性码C的生成矩阵得出其对偶码C⊥及Gray象Φ(C)的生成矩阵,证明了F2+uF2+u2F2上线性码的Gray象及其对偶码的Gray象互为对偶码,并给出了F2+uF2+u2F2上线性码自对偶的一个充要条件。

This paper defines a new Gray map from the ring F2＋uF2＋u^2F2 to F2, then gets the gener-ator matrix of the dual code C⊥ and Gray image Ф（C） from the generator matrix of the linear code C o-ver F2＋uF2＋u^2F2, and proves that the dual code of the Gray image Ф（C） is Ф（C⊥）. And a necessary and sufficient condition of the linear code over F2＋uF2＋u^2F2 which is self-dual is given.