A now deleted answer put that down to an argument based on amino-acid coding in codons but, as pointed out past Konrad Rudolph in a reaction, arguments based on codons cannot be correct because the 4-base system likely pre-dates the advancement of protein translation. I urge that it comes from a combination of factors which variety it a 'just right' suited for RNA-based replicators: The four bases are able to impression with high reliability because two features combine to exclude mis-pairings: More bases must, inevitably, let up the strength of these exclusionary approaches and increase the sign of mispairings.
RNA world replicators will have lacked the complex repair and error checking mechanisms of modern lifeforms so that increase in mispairings would reasonable not be corrected. Of without a doubt, a two base system would be even more capable of reducing errors. I suggest so as to the advantage of the four base system is that it allows for much more intricate 3-dimensional structure to form modish RNA and thus enabled a broader range of catalytic capabilities.
I've only considered even-numbers since only even numbered systems could use the Watson-Crick style of base pairing.
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