On Even Length Codes Over Finite Rings

Codes over finite rings have been studied in the early 1970. A great deal of attention has been given to codes over finite rings from 1990, because of their new role in algebraic coding theory and their successful application. The key to describing the structure of cyclic codes over a ring R is to view cyclic codes as ideals in the polynomial ring R[x]/, where n is the length of the code. In previous studies, some authors determined the structure of cyclic codes over Z4 for arbitrary even length by finding the generator polynomial, the number of cyclic codes for a given length and the duals for these codes, and also determined the structure of negacyclic codes of even length over the ring Z2a and their dual codes. In this thesis, we introduce cyclic codes of an arbitrary length n over the rings F2 + uF2 with u2 = 0 mod 2 and F2 + uF2 + u2F2 with u3 = 0 mod 2. We find a set of generators for these codes. The rank and the dual of these codes are studied as well. We will extend these results about the rings F2 +uF2 and F2 +uF2 +u2F2 to more general rings F2 + uF2 + u2F2 = : : : + uk-1F2 with uk = 0 mod 2. Finally we study the structure of (1 + u)-constacyclic codes of even length n over the ring F2 + uF2 with u2 = 0 mod 2. Also we extend this study to the ring F2 + uF2 + u2F2 with u3 = 0 mod 2