环F2＋uF2上长为2^e的循环码

Cyclic Codes of Length 2^e Over F2＋uF2

近十多年来,有限环上的循环码一直是编码研究者所关心的热点问题,本文证明了R[x]/<xn?1>不是主理想环,其中R=F2+uF2,u2=0且n=2e。分3种情形讨论了环R[x]/<xn?1>中的非零理想,并给出了R上循环码的可以唯一确定的生成元的表达形式,同时给出了R上循环码的李距离的一个上界估计。

In the last ten more years, cyclic codes over finite rings have become a hot issue for coding theorists.It is proved that R[x]/〈x^n- 1〉 is not a principal ideal domain, where R=F2 ＋ uF2 with u2=0, and n=2^e. The nonzero ideals of R[x]/〈x^n- 1〉 are discussed in three cases and the expressions of the uniquely determined generators of the cyclic codes are given. An estimate of upper bound of Lee distance of cyclic codes over R is also given.