In this article, we examine a particular family of infinite-dimensional discrete autonomous systems given by a first-order state-space equation; the state transition matrix for this family is a Laurent polynomial matrix A(sigma, sigma(-1)), where s is the shift operator on R-n-valued sequences. We term this family of systems as Laurent systems. We give necessary and sufficient conditions for the exponential l(2)-stability and the exponential l(infinity)-stability of Laurent systems. We also compare the following four different notions of stability for Laurent systems: the l(2)-stability, the exponential l(2)-stability, the l(infinity)-stability, and the exponential l(infinity)-stability; furthermore, we conclude that the l(2)-stability is an outlier.