F_3 ve F_5 üzerindeki self-dual kodların ağırlıkenumeratörleri için yapısal teorem

For codes over a prime field Fp, the Lee weight enumerators of linear codes over Fp are known to be quite accessible, useful and, interesting. This is partly due to the fact that they are intimately connected to the theory of Hilbert modular forms over the totally real subfield of the p-th cyclotomic field. The aim of this master thesis is to reprove two theorems that describe explicitly the ring generated by the Lee weight enumerators of self-dual codes over F_3 and F_5, respectively. The main point of this project is that the proofs given in this thesis differ from the ones that can be found in the existing literature. For example, [Ebe02] proves the case of F_5 using the theory of Hilbert modular forms over the quadratic field. In contrast to that approach, the proofs in this thesis are purely algebraic based on the theory of Weil representations of SL(2;Z) and follow ideas of Boylan and Skoruppa explained in their Coding Theory lecture notes [BS16]. This thesis explains in detail the methods and ideas used in [BS16] for deducing the structure of the ring generated by the Lee weight enumerators of self-dual codes over F_3 and applies these methods to give a new proof for the explicit description of the Lee weight enumerators of self-dual codes over F_5.