Research On Several Classes Of Constacyclic Codes Over Finite Rings

With a more deeply study in error-correcting codes over finite chain rings,constacyclic codes over some finite rings which are not finite chain rings also havebeen considered by some authors. In this paper, we mainly study the structures ofseveral classes of constacyclic codes over finite rings（not finite chain rings） andalso the properties of their images under the Gray maps. The details are given asfollows:1. We introduce a Gray map from F_p+vF_p to F_p²and study （1-2v）-constacycliccodes overF_p+vF_p, wherev²=v. It is proved that the image of a （1-2v）-constacycliccode of length n over F_p+vF_p under the Gray map is a distance-invariant linear cycliccode of length2n over F_p. The generator polynomials of such constacyclic codes for anarbitrary length and their images are determined, their dual codes are also discussed.2. We study （1+u）-constacyclic codes over F₂+uF₂+vF₂+uvF₂. Based on the Leedistance and homogeneous distance we give two Gray maps. It is proved that the image of（1+u）-constacyclic codes over F₂+uF₂+vF₂+uvF₂of length n under the former Graymap is a distance-invariant binary quasi-cyclic codes of index2and length4n whileunder the later Gray map is a distance-invariant binary quasi-cyclic codes of index4andlength8n. The generators of such constacyclic codes for an arbitrary length aredetermined, and some optimal binary quasi-cyclic codes are also obtained.3.（1+u）-constacyclic codes over F_p+uF_p+vF_p+uvF_p are studied. The structure ofsuch constacyclic codes for an arbitrary length is determined. By using this structure, somep-ary codes with good parameters are also constructed.