From M-W, definition (5): 5: having more than one codon representing an amino acid; also : being such a codon this is, of course, not that helpful. It is terminology borrowed from mathematics: 2: being mathematically simpler (as by having a factor or constant equal to zero) than the typical case <a degenerate hyperbola> Basically, it is specialized terminology. If more than one codon can code for an amino acid, it is degenerate because there is not a one-to-one correspondence between codons and amino acids.
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