Research On Trace Codes And Constacyclic Codes Over Several Finite Rings

With the rapid development of error-correcting over finite fields,the cod-ing theory over finite rings has also been paid much attention by the researcher-s.Based on the previous excellent contributions on coding theory over finite rings,we construct a series of linear codes and determine their Lee weight dis-tribution.In addition,the constacyclic codes over finite rings are considered.The details are given as follows:1、We extend the concept of trace code over finite fields to the finite chain ring R1 = Fp+uFp with u2 = 0.When p = 2,there exist a multiplica.tive homomorphism from the ring R1 to the ring Z4.We construct a class of linear codes.Here we call them trace codes,the defining set of these codes consists of the unit group.Applying the linear Gray map,different from the ring Z4,we get a class of binary two-weight linear codes.When p is an odd prime number,the defining set of the trace code is only a subset of the unit group but no longer a subgroup.Due to characters of finite fields,we determine the Lee weight distribution of the trace code.Using the Gray map,under some conditions,we get a class of optimal two-weight linear codes over Fp.2.We extend the concept of trace code over finite fields to the finite non-chain ring R2 = F2 + F2 =vF2 + v3F2 + v4F2 with v5=1.For a given integer m ∈ N,according to the Chinese Remainder Theorem,the value of gcd(m,4)directly affects the order of the unit group of ring R2(m)Combined with the linear Gray map,we obtain three families of binary linear codes with different weights.3.Several kinds of special linear codes over the finite field Fp are in-vestigated.Combining exponential sums over finite fields and Pless Power Moments,we discuss the weight distributions of these types of linear codes.In addition,we also give their complete weight enumerators in the case of p|mt.4、We study constacyelie codes over the finite· non-chain ring Fq2 +vFq2 and repeated-root const acyclic codes over the finite field Fq.After in-depth study of the constacyclic codes,we give a method of construction of MDS constacyclic codes.For repeated-root constacyclic codes over Fq we consider the two cases where the code length is 3lm ps and klps.