  We use the general notion of set of indices to construct algebras of
nonlinear generalized functions of Colombeau type. They are formally defined in
the same way as the special Colombeau algebra, but based on more general growth
condition formalized by the notion of asymptotic gauge. This generalization
includes the special, full and nonstandard analysis based Colombeau type
algebras in a unique framework. We compare Colombeau algebras generated by
asymptotic gauges with other analogous construction, and we study
systematically their properties, with particular attention to the existence and
definition of embeddings of distributions. We finally prove that, in our
framework, for every linear homogeneous ODE with generalized coefficients there
exists a minimal Colombeau algebra generated by asymptotic gauges in which the
ODE can be uniquely solved. This marks a main difference with the Colombeau
special algebra, where only linear homogeneous ODEs satisfying some restriction
on the coefficients can be solved.
