环Fq＋uFq＋…＋u^k-1Fq上一类重根常循环码

A Class of Repeated-root Constacyclic Codes over the Ring F_q+uF_q+…+u^(k-1)F_q

记R=F_q+uF_q+…+u^(k-1)F_q,G=R[x]/<x^t-α>,且λ是R中可逆元。定义了从G^n到R^tn的新的Gray映射φ,证明了J是G上长为n的线性的x-常循环码当且仅当φ(J)是R上长为tn的线性的α-常循环码。使用有限环理论,获得了环R上长为p^e的所有的(uλ-1)-常循环码的结构及其码字个数。特别地,获得了环F2m+uF上长为2~e的(uλ-1)-常循环码的对偶码的结构及其码字个数。推广了环2aZ上重根负循环码的若干结果。

Let R=Fq＋uFq＋…＋uk-1Fq,G=R[x]/〈 x^t -α 〉, and A be an invertible element in R. A new Gray map φ from G^n to R^tn is defined. It is proved that J is a linear x -constacyclic code of length n if and only if φ（J） is a linear α-constacyclic code over R of length tn. By means of the theory of finite rings the structure and sizes of all （uλ- 1） -constacyclic codes over R of length p^e are obtained. Especially, the structure and sizes of the duals of all （uλ - 1） -constacyclic codes of length 2^e over the ring F2m＋uF2m are also obtained. Some of the results about repeated-root negacyclic codes over the ring Z2h are generalized.