Why the standard genetic code has its particular structure
In plain words
All life reads three-letter DNA words with nearly the same dictionary, which assigns 64 words to 20 amino acids. That dictionary limits the damage from copying errors better than almost any random dictionary, and whether this came from chemistry, history or selection is unknown.
Precise statement
Under a cost function based on amino-acid polar requirement weighted by point mutation and mistranslation, approximately one random code in $10^6$ with the same block structure outperforms the standard code (Freeland and Hurst, 1998). Determine which mechanism produced this structure: frozen accident, stereochemical codon or anticodon affinity, coevolution with amino-acid biosynthesis, or selection for error minimization, with a quantitative test of whether the observed optimization exceeds what each non-adaptive mechanism predicts.
What would settle it
Computation of the error-cost distribution generated by each candidate mechanism, combined with measured codon or anticodon affinities, showing which reproduces the observed optimization level and codon assignments.
Status in the literature
A review (Koonin and Novozhilov, IUBMB Life 2009, doi 10.1002/iub.146) concludes that several factors combined; no single mechanism is established.