Modules with Non-Cyclic Socle and the Extension Property

Abstract

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.