环F_2+uF_2的Galois扩张上的迹码

Trace Codes over Galois Extensions of Ring F_2+uF_2

环F2+uF2是介于环Z4与域F4之间的一种四元素环,因此分享了环Z4与域F4的一些好的性质,此环上的编码理论研究成为一个新的热点。该文给出了环F2+uF2的Galois扩张的相关理论,指出此Galois扩环的自同构群不同于Z4环上的Galois扩环的自同构群;定义了Galois扩环上的迹码的概念及子环子码的概念,证明了此Galois扩环上的一个码的对偶码的迹码是该环的子环子码的对偶码。

F2＋uF2 is a ring with four elements which shares some good properties of both Z4 and F4. Coding theory over this ring has recently received a great deal of interest among coding theorists. This paper gives the theory of Galois extensions over F2 ＋uF2, and shows that the automorphism groups of these Galois extensions are different from the corresponding groups over Z4. Trace codes and subring subcodes over Galois extensions are defined, and it is proved that the trace codes of dual codes of linear codes are the dual codes of subring subcodes.