The Study Of Coding Over Galois Ring GR(q～m)

In this correspondence, we study codes over Galois ring GR (q^m) on the base of the theory of error-correcting codes and quaternary codes.Let q= p^t where p is a prime integer and t is a positive integer. Let n is a positive integer and (n, p) =1. Ineger ring Z mod k forms a residue ring Z_k.At first, we define a ring homomorphism from Z_q to Z_p, and prove that the polynomial xⁿ-1 over Z_q can be uniquely factored into a product of finitely many pair-wise coprime basic irreducible polynomials over Z_q. And then, we give Hensel's lemma and Hensel lift over Z_q.The second, we define a Frobenius map f of GR (q^m) and the trace map over GR (q^m) , simultaneously give the unique representation of any element of GR (q^m) and some properties of these maps. Using the trace map's definition and linear property, we prove that the trace map has transitivity.In the finally, we give any ideal of ring GR (q^m) [x]/ (xⁿ-1) to be a sum of (f_i (x)), (pf, (x)), ……, and (p^t-1 f, (x)), where f_i (x) = (xⁿ-1) /f_i, (x), 1≤i≤r, and the trace representation of cyclic codes C over GR (q^m), i. e.