Research On Some Codes Over Finite Chain Rings

As the development of error-correcting codes, classical coding theory takes place in the setting of vector spaces over finite fields. In the 1990's, 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 Z 4under the Gray map. Since then, codes over finite rings have been given more attention, and codes over finite chain rings become a hot topic. In this paper, we mainly study some codes and properties over several finite chain rings, details are given as follows:1.Use a new Gray map to give the generator matrix of a linear code and its Gray image, then prove that the Gray image of C is the dual code ofΦ(C⊥), in whichΦ(C⊥)is the Gray image of C⊥.Then give a necessary and sufficient condition of linear code which is self-dual.2. Establish a relation between the support weight distribution of a linear code of type8k and the Hamming weight distribution over ring F₂ + uF₂ + u²F₂, and derive a relation between the support weight distribution of the linear code and its dual by MacWilliams-identity over the ring.3. We discuss the self-dual codes and Typeâ…¡codes over the finite chain ring F₂ m + uF₂ m + + u kF₂m. Properties and relationships among several important Gray maps over the ring are given, we give the necessary and sufficient condition for the existence of Typeâ…¡codes of length n over the ring, and get a conclusion that the minimum Lee weight of self-dual codes of length n does not exceed 2n .4. At last, we give a simple method to construct special cyclic code over Fq and discuss the construction and property.