Research On Some Issues In Coding Theory Over Rings In Management Information

Management informatics is an integration of management science and information technology. Information transformission and information coding are both the important contents of management informatics. Error-correcting codes over finite fields have played important roles in information transformission and information coding through more than sixty years of development. Recently, People have fully realized that researches on error-correcting codes over finite rings have not only important theoretical significancance but also certain practical significancance because of the extensive links between codes over finite rings and codes over finite fields through many different Gray maps.This dissertation focuses on cyclic codes and constacyclic codes over the residue class rings Fq+uFq+…+uk-1Fq of many kinds of lengths because these rings have a variety of applications in coding theory. The details are given as follows.1. We prove that the simple-root cyclic codes over Fp+uFp+…+uk-1Fp are principally generated and give sufficient and necessary conditions for these codes to be self-dual. We also give a sufficient and necessary condition for these codes to be free modules and determine the idempotent generators of these codes and their dual codes.2. We obtain the structure of cyclic codes over F2+uF2of length2e. We prove that R[x]/<xn-1> is not a principal ideal ring, where R=F2+uF2and n=2e. We obtain the expression form of the uniquely determined generators of cyclic codes over F2+uF2of this length in three kinds of situations according to whether these codes contains monic polynomials (not according to whether these codes are principal ideals). We also give a sufficient and necessary condition for the third-case cyclic codes to be principally generated and obtain an upper bound on Lee distance about cyclic codes over F2+uF2of length2e.3. We obtain the direct decomposition of (1+u)-onstacyclic codes over F2+uF2of arbitrary lengths and give a sufficient and necessary condition for these codes to be self-dual. We prove that self-dual (1+u)-constacyclic codes over F2+uF2are of Type Ⅰ, but not of Type Ⅳ. Finally, we discuss the Euclidean distance of self-dual (1+u)-constacyclic codes over F2+uFZ.4. We define the covering radius of codes over F2+uF2with respect to Lee distance and obtain several upper and lower bound on the covering radius of codes over F2+uF2for Lee distance by means of Lee weight and linear Gray map.5. We study cyclic codes of arbitrary lengths over Fq+uFq and their dual codes. By means of the theory of ring homomorphism we obtain the expression form of the uniquely determined generators of these cyclic codes. We also determine the minimal generating sets and the ranks of these cyclic codes. At last, we obtain the generators of the dual codes of these cyclic codes.6. We investigate (uλ-1)-constacyclic codes of arbitrary lengths over R=Fq+uFq+…+uk-1Fq, where λ is any invertible element of R. Firstly, we obtain the structure and sizes of all (uλ-1)-constacyclic codes over R of length pe by means of the theory of finite rings. Especially, the structure and sizes of the duals of all (uλ-1)-constacyclic codes of length2e over the ring F2m+uF2a are also obtained. Secondly, we obtain the structure of all (uλ-1)-constacyclic codes over R of an arbitray length N by using the theory of ring homomorphism and prove that R[x]/<xN+1-uλ> is a principal ideal ring. We also give the number of the ideals of R[x]/<xN+1-uλ> and generators of these ideals. Finally, we obtain the generator polynomials of the highest-order torsion codes of all these (uλ-1)-constacyclic codes of an arbitrary length over R. As a result, the Hamming distance of all these (uλ-1)-constacyclic codes of length pe is obtained.7. We prove that every ternary cyclic code of length3m of some kind is the Nechaev-Gray map of certain linear code of length n over the ring F3+uF3by defining the new generalized Nechaev permutation and the new Gray map over the ring F3+uF3. The new Gray map can induce Van-Lint’s generalized (U|U+V)-construction. A detailed proof of the distance formula about the construction is also given.