  We study the question of the existence of a Waldhausen category on any
(relative) abelian category in which the contractible objects are the
(relatively) projective objects. The associated $K$-theory groups are "stable
algebraic $G$-theory," which in degree zero form a certain stable
representation group. We prove both some existence and nonexistence results
about such Waldhausen category structures, including the fact that, while it
was known that the category of $R$-modules admits a model category structure if
$R$ is quasi-Frobenius, that assumption is required even to get a Waldhausen
category structure with cylinder functor---i.e., Waldhausen categories do not
offer a more general framework than model categories for studying stable
representation theory of rings. We study multiplicative structures on these
Waldhausen categories, and we relate stable algebraic $G$-theory to algebraic
$K$-theory and we compute stable algebraic $G$-theory for finite-dimensional
quasi-Frobenius nilpotent extensions of finite fields. Finally, we show that
the connective stable $G$-theory spectrum of $\mathbb{F}_{p^n}[x]/x^{p^n}$ is a
complex oriented ring spectrum, partially answering a question of J. Morava
about complex orientations on algebraic $K$-theory spectra.
