Isometry Groups of Additive Codes

When C ⊆ F is a linear code over a finite field F, every linear Hamming isometry of C to itself is the restriction of a linear Hamming isometry of F to itself, i.e., a monomial transformation. This is no longer the case for additive codes over non-prime fields. Every monomial transformation mapping C to itself is an additive Hamming isometry, but there exist additive Hamming isometries that are not monomial transformations. The monomial transformations mapping C to itself form a group rM(C), and the additive Hamming isometries form a larger group Isom(C): rM(C) ⊆ Isom(C). The main result says that these two subgroups can be as different as possible: for any two subgroups H1 ⊆ H2, subject to some natural necessary conditions, there exists an additive code C such that rM(C) = H1 and Isom(C) = H2.