Research On Structure Properties Of Linear Codes Over Finite Rings

Classical coding theory takes place in the setting of vector spaces over finite fields.In the 1980s,the theory of error-correcting codes over finite rings has experienced tremendous growth since the significant discovery that several well-known prominent families of good nonlinear binary codes can be identified as images of linear codes over Z4 under the Gray map.Since then,codes over finite rings have been given more attention.In this paper,we study the structure of the (1+u)-constacyclic codes of an arbitrary length and their duals over R1=F2+uF2+u2F2. Further, we investigate some structure properties of (1-u)-cyclic codes over R2=Fpk+uFpk.A necessary and sufficient condition for the existence of self-dual(1-u)-cyclic codes over F2k+uF2k is obtained.The details are given as follows:(1)Using a homomorphism from R1[x]/〈xn-(1+u)〉to F2[x]/〈xn-1〉,we obtain the generators of(1+u)-constacyclic codes of an arbitrary length over the polynomial residue ring F2+uF2+u2F2,and give a classification of such (1+u)-constacyclic codes.(2) The rank of (1+u)-constacyclic codes over F2+uF2+u2F2 of length n=2em(m odd)is obtained,and the basis of the minimal spanning sets of such codes are found.Furthermore,it is proved that the dual of a(1+u)-constacyclic code is a principal ideal of R1[x]/(xn-(1+u)).(3)Using an isomorphism from R2[x]/〈xn-1〉to R2[x]/〈xn-(1-u)〉,we prove that any(1+u)-cyclic codes over R2=Fpk+uFpk is a principal ideal.The generators and the number of these codes are determined.(4)we derive the generators of the duals of (1-u)-cyclic codes over F2k+uF2k and give a necessary and sufficient condition for the existence of self-dual codes.