Isometry groups of additive codes over finite fields

Abstract

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