The genetic code is the function from the set of codons to the set of amino acids by which a DNA sequence encodes proteins. Since the codons also influence the shape of the DNA molecule itself, the same sequence that encodes a protein also has a separate geometric interpretation. A question then arises: How well-duplexed are these two "codes"? In other words, in choosing a genetic sequence to encode a particular protein, how much freedom does one still have to vary the geometry (or vice versa). A recent paper by the first author addressed this question using two different methods. After reviewing those results, this paper addresses the same question with a third method: the use of Monte Carlo and Gaussian sampling methods to approximate a multi-integral representing the mutual information of a variety of possible genetic codes. Once again, it is found that the genetic code used in nuclear DNA has a slightly lower than average duplexing efficiency as compared with other hypothetical genetic codes. A concluding section discusses the significance of these surprising results.