| In resent years, the theory of ring of coding and cryptography is the key for researcher in coding and cryptography. In particular, the study of code over Z4. In this paper, the main results are the following:1. We define a Gray map over Zp[u]/(um-1).and the Gray map is an isometry from (Zp[u]/(um-1), dG) to (Zpmn, dH).Moreover, if G is a generator matrix of a code C over Zp[u]/(um-1),then a generator matrix of (C) can be obtained. A necessary and sufficient condition for the Gray image of cyclic code to be quasi-cyclic is given.2. Let Z4 be the integers modulo 4, the notion of cyclic codes over Z4 is introduced. We provide a trace description of such a code.3. A ring isomorphism between linear (1 -pk) - cyclic codes and linear (1 -tpk) -cyclic codes over Z pk-1 is constructed in this paper. We study the (1 - tpk) -cyclic codes and product the generator of (1 - tpk) -cyclic codes over Z pk-1 .4. We define a Frobenius map over Galois, trace codes and subring subcode. We prove the trace of dual codes over Galois is the dual codes of subring subcode.All the results are very important for the study of the error correcting codes over rings. |
