Before revision:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension  k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em   .
After revision:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension k on k^{O(1)} points in k^{O(k)} operations. The second design generalizes the Bj\"orklund~{\em .