The Extension Theorem with Respect to Symmetrized Weight Compositions

We will say that an alphabet A satisfies the extension property with respect to a weight w if every linear isomorphism between two linear codes in A that preserves w extends to a monomial transformation of A. In the 1960s MacWilliams proved that finite fields have the extension property with respect to Hamming weight. It is known that a module A has the extension property with respect to Hamming weight or a homogeneous weight if and only if A is pseudo-injective and embeds into O R. The main theorem presented in this paper gives a sufficient condition for an alphabet to have the extension property with respect to symmetrized weight compositions. It has already been proven that a Frobenius bimodule has the extension property with respect to symmetrized weight compositions. This result follows from the main theorem.