Modules with Non-Cyclic Socle and the Extension Property

In 2009, J. Wood proved that Frobenius bimodules have the extension property for symmetrized weight compositions. More generally, it was later shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. In this thesis, a partial converse is proved. For a significant class of finite module alphabets, the cyclic socle condition is shown necessary for satisfying the extension property. The idea is to use a new weight function to return to the original case of Hamming weight.