We study the dynamics of genetic code evolution. The model of Vetsigian et al. [Proc. Natl. Acad. Sci. USA 103, 10696 (2006)PNASA60027-842410.1073/pnas.0603780103] and Vetsigian [Collective evolution of biological and physical systems, Ph.D. thesis, 2005] uses the mechanism of horizontal gene transfer to demonstrate convergence of the genetic code to a near universal solution. We reproduce and analyze the algorithm as a dynamical system. All the parameters used in the model are varied to assess their impact on convergence and optimality score. We show that by allowing specific parameters to vary with time, the solution exhibits attractor dynamics. Finally, we study automorphisms of the genetic code arising due to this model. We use this to examine the scaling of the solutions to re-examine universality and find that there is a direct link to mutation rate.