Infinite Sequence And Cyclic Codes Over Finite Chain Ring

In this paper we give a method to compute the period and depth of infinite sequence, with this method, we can find the depth of infinite sequence on Z_c, where c = p₁^m₁p₂^m₂…p_n^m_n, from it's period; and we can find the period on Z_c from the depth of infinite on each Z_pim_i where i = 1,2,…, n. And we give the generators of any cyclic code over a finite chain ring. We showed that for any integer k with 1≤k≤m(c, n), there exists a cyclic code C≤Z_c[x]/n—1> which is generated by k polynomials but cannot be generated by k—1 polynomials. if R is a finite chain ring of charactristic p^α, let n—p^βl with p (?)l. m = min{α,p^β}. Then for any integer k with 1≤k≤m, there exists a cyclic code C of R[x]/n—1> with is generated by k polynomials but cannot generated by k—1 polynomials, if R is a finitate chain ring of charactristic p^α, and p (?)n. Then R_n = R[x]/(xⁿ—1) is a principal ring.