Research On Constacyclic Codes Over Some Classes Of Finite Rings

In this dissertation, the properties of constacyclic codes over finite rings are studied. This dissertation contains three main parts:(1) We study Hermitian self-orthogonal constacyclic codes over the ring F2+uFq2. Firstly, it is shown that the Hermitian dual code of a λ-constacyclic codes over Fq2+uFq2 is a λ-q-constacyclic code. Secondly, the generator polynomial of constacyclic codes over Fq2+uFq2 is given. Finally, on this basis, a sufficient and necessary condition for the existence of self-orthogonal constacyclic codes is determined. Furthermore, the enumerator formula of self-orthogonal constacyclic codes is obtained.(2) We study constacyclic codes over the finite non-chain ring Fq+uFq+vFq, where u2= v2= 0, uv= vu= 0. The structure of (1-u-v)-constacyclic codes of an arbitrary length over the ring Fq+uFg+vFg is determined by using a homomorphism. A Gray map from Fq+uFq+vFq to Fp2p is introduced. It is shown that the Gray images of (1-u-v)-constacyclic codes over Fq+uFq+vFq of length n are linear quasi-cyclic codes of length 2pn and index 2 overFp. By using the Gray map, some optimal linear codes over finite fields are obtained. (3) We study the depth spectrum of negacyclic codes over Z4 of evenlength. By by using its residue and torsion codes, the depth spectrum of any negacyclic code over Z4 of even length is completely determined.