Researches On The Cyclic Code And A Class Of Constacyclic Codes Over Finite Rings

Error-correcting coding theory is the main theoretical basis for reliability of information transmission. As the technique of encoding and decoding develops, error-correcting coding theory has been widely applied and studied. Gradually, the theoretical and practical value of error-correcting codes over finite rings was discovered by the researchers. Cyclic codes, constacyclic codes, negacyclic codes and self-dual codes over finite rings have become a focus of the study of error-correcting codes.This paper mainly studies the structure of cyclic codes and constacyclic codes over finite rings. Firstly we study the property of torsion codes over finite chain ring F2+uF2+u2F2, and discuss the structure of the generator polynomials of the cyclic code. It is proved that the residue code and torsion code that generate the cyclic code over the ring are binary cyclic codes as well. Secondly we find out the generator polynomials of the dual code, and discuss the necessary and sufficient conditions of the existence of the self-dual cyclic code over ring F2+uF2+u2F2. Finally we define a Gray map from (F2+uF2+u2F2)n to F23", and prove that the Gray distance of the cyclic code is equal to the minimum value between the Hamming distance of the1st torsion code and three times of the Hamming distance of the2nd torsion code.On the other hand, we study the structure of (1+Am)-constacyclic code over finite ring Fq+uFq+vFq+uvFq, q is the power of the prime p. We define a Gray map from (Fq+uFq+vFq+uvFq)n with homogeneous distance to Fq4nwith Hamming distance, and give the relationship among the homogeneous distance of (1+λu)-constacyclic code, the Hamming distance of the residue code and the torsion code.