Submodule codes over some rings

Since the latter half of the century, the digital communication has been dramatically developed, and caused transmission and storing data to gain more importance. Hence algebraic coding theory which aims preventing or controlling the data loss arised and developed rapidly. Cyclic codes, which are a significant class of linear codes, are vital for the communication theory, because of their mathematical properties and the natural adaption of these properties to electronic circuit structures and linear feedback shift registers (LFSR). Although some non-linear codes may have better parameters, capability of error correcting for instance, they are more difficult to find and study since they are not as systematic as linear codes. However, in 1994, Hammons et. al. presented how to obtain more acceptable non-linear codes via considering linear codes by showing the relation between quaternary linear codes and non-linear binary codes. After this work which inspired many works, the structure of cyclic codes and the Gray images of these codes have been studied over various rings. Recently, cyclic codes over Zq+uZq where q is a prime power and u^2=0 have been also considered. However, these studies do not deal with the ideal structure of the ring (Zq+uZq)[x]/ in spite of the fact that a linear code C of length n over a ring R is cyclic if and only if the corresponding polynomials of its codewords form an ideal structure in the quotient ring R[x]/ . In this thesis, we determine all the ideals of the rings Zq+uZq, where q=p^s, p is any prime and s is any positive integer with u^2=0. Next, we give a formula that enumerates the number of ideals of these rings. Afterwards, we investigate the cyclic codes of length n, where n is relatively prime to q, considering the ideal structure of the quotient ring (Zq+uZq)[x]/. In this way, we determine the number of these ideals by giving a formula. We consider some special family of cyclic codes for some fixed values of q and we find the size of these codes. We also consider dual codes of these special families and give the size of them.