| Linear codes play an important role in both coding theory and decode theory.In this dissertation, we investigate the properties of linear codes over different residue class ring, the main results are as follows:1. The relation between cyclic codes(negacyclic, quasi-cyclic codes) over Zm(m≥6) and cyclic codes(negacyclic, quasi-cyclic codes) over theirs direct summand is studied by making use of the ring direct summand decomposition. Moreover, the relation between codes over Zm and the rings which have the same characteristics of Zm's direct summand is also investigated by defining the isomorphism mappingψof Zm to the rings which have the same characteristics of Zm's direct summand. Meanwhile, applying the definition of the Gray mapping (?):Zpk+1→Zp,the generally relation betweenλ-cyclic codes over Zm and the quasi-cyclic codes over rings which have the same characteristics of Zm's direct summand is given. At the same time, necessary and sufficient conditions for the existence of self-dual codes over Zm are obtained.2. Applying the concept 0 of Gray map, the relation between cyclic codes over Zpk+1n and the quasi-cyclic codes of index pk-1 and length pkn over Fppkn is gavin. At the same time, we introduce mapψon Zp2n→Fp2n and study the relation between negacyclic codes and cyclic codes .3. The concept of double chain rings is given at first. We investigate negacyclic codes and cyclic codes of length ps over the ring Fpk+uFpk,prove that negacyclic codes of length ps are precisely the ideals of the double chain ring (Fpk+uFpk)[x]/(xps+1),and give their Hamming distance distributions on the double chains by the charactrics of chain rings. We find that there exist self-dual codes if and only if p=2.At the same time, we obtain the Hamming distance distributions of cyclic codes on the double chains with length ps. |
