Extension theorems for various weight functions over Frobenius bimodules

In this paper, we study codes where the alphabet is a finite Frobenius bimodule over a finite ring. We discuss the extension property for various weight functions. Employing an entirely character-theoretic approach and a duality theory for partitions on Frobenius bimodules, we derive alternative proofs for the facts that the Hamming weight and the homogeneous weight satisfy the extension property. We also use the same techniques to derive the extension property for other weights, such as the Rosenbloom–Tsfasman weight.