F2+vF2 ve F2xF2+vF2 halkaları üzerine self-dual kodların inşaası

In this dissertation, self-dual codes over the rings F2+vF2 and F2xF2+vF2 with v^2=v are considered. The necessary and sufficient conditions in order to obtain Euclidean and Hermitian self-dual codes over both rings are given. By using the distance preserving Gray maps, codes over these rings are transferred to the binary field F2. By using some circulant and some symmetric matrices, free Euclidean and free Hermitian self-dual codes over the ring F2+vF2 of even length are obtained. A new shortening method which enables us to obtain Hermitian self-dual codes of odd length by using Hermitian self-dual codes of even length is presented. The Hermitian self-dual code of length 31 with minimum Hamming weight 8 whose existence was not known previously is found by using the shortening method. The structure of the ring F2xF2+vF2, which is a commutative non-chain ring of characteristic 2, is investigated. Linear codes over this ring are defined. By using some circulant and some symmetric matrices, free Euclidean and free Hermitian self-dual codes over the ring F2xF2+vF2 of even length are found. The complete, symmetrized, Hamming and Lee weight enumerators for the linear codes over the ring F2xF2+vF2 are defined and the MacWilliams identities are proved.