From what I understand (see, e.g., this question), the Monster Moonshine Module is a kind if "third generation" (or "second quantization"?) after the Golay code (with automorphism group $M_{24}$) and the Leech lattice (with automorphism group $\mathit{Co}_0$).

Now the Golay code has applications besides constructing the Mathieu groups: it has practical applications as an error-correcting code (e.g., for sending the Voyager probes' photos of the giant planets of the solar system). The Leech lattice also has applications besides constructing the Conway groups: it provides an efficient analog-to-digital quantizer. There are interesting engineering/algorithmic questions on how to efficiently perform maximal likelihood decoding of the Golay code or Leech lattice (for a readable account of the latter, see, e.g., chapter 4 in Alex van Poppelen's masters thesis, "Cryptographic decoding of the Leech lattice").

This suggests the rather wild idea:

Is there any known, or simply conceivable, sort of engineering or practical application of the Moonshine module?

(Of course, one could ask this question for VOA's in general and not just for this particular one. But without a clear idea as to what applications might be possible, it seems sensible to concentrate on this exceptional object whose aforementioned "progenitors" seem to have very good properties along those lines so maybe it is reasonable to hope that it too could have some. Also note that I'm not asking about applications in theoretical physics unless it is at the intersection with information theory in some sense.)

Now maybe this is a bit too wildly speculative, so I'll asked a somewhat toned down question:

Has the Moonshine module been studied from the algorithmic perspective?

An even more toned down question would be whether somewhere in the literature there exists a description of it that is targeted toward computer implementation, or an actual computer implementation (that can, among other things, efficiently compute a specified number of coefficients of the product of two elements). These kinds of descriptions (as well as implementations) exist for the Golay code and the Leech lattice.

(I gather from its absence in Wilson's paper *New Computations in the Monster* that the Moonshine module is not algorithmically useful toward a computer implementation of elements of the Monster group. In a sense, this is disappointing. But this does not mean that it is not good for anything.)