The genetic code maps the 64 nucleotide triplets (codons) to 20 amino acids. Some argue that the specific form of the code with its 20 amino acids might be a 'frozen accident' because of the overwhelming effects of any further change. Others see it as a consequence of primordial biochemical pathways and their evolution. Here we examine a scenario in which evolution drives the emergence of a genetic code by selecting for an amino acid map that minimizes the impact of errors. We treat the stochastic mapping of codons to amino acids as a noisy information channel with a natural fitness measure. Organisms compete by the fitness of their codes and, as a result, a genetic code emerges at a supercritical transition in the noisy channel, when the mapping of codons to amino acids becomes non-random. At the phase transition, a small expansion is valid and the emergent code is governed by smooth modes of the Laplacian of errors. These modes are in turn governed by the topology of the error-graph, in which codons are connected if they are likely to be confused. This topology sets an upper bound-which is related to the classical map-coloring problem-on the number of possible amino acids. The suggested scenario is generic and may describe a mechanism for the formation of other error-prone biological codes, such as the recognition of DNA sites by proteins in the transcription regulatory network.