  A code over GF$(q^m)$ can be imaged or expanded into a code over GF$(q)$
using a basis for the extension field over the base field. The properties of
such an image depend on the original code and the basis chosen for imaging.
Problems relating the properties of a code and its image with respect to a
basis have been of great interest in the field of coding theory. In this work,
a generalized version of the problem of self-orthogonality of the $q$-ary image
of a $q^m$-ary code has been considered. Given an inner product (more
generally, a biadditive form), necessary and sufficient conditions have been
derived for a code over a field extension and an expansion basis so that an
image of that code is self-orthogonal. The conditions require that the original
code be self-orthogonal with respect to several related biadditive forms
whenever certain power sums of the dual basis elements do not vanish. Numerous
interesting corollaries have been derived by specializing the general
conditions. An interesting result for the canonical or regular inner product in
fields of characteristic two is that only self-orthogonal codes result in
self-orthogonal images. Another result is that image of a code is
self-orthogonal for all bases if and only if trace of the code is
self-orthogonal, except for the case of binary images of 4-ary codes. The
conditions are particularly simple to state and apply for cyclic codes. To
illustrate a possible application, new quantum error-correcting codes have been
constructed with larger minimum distance than previously known.
