At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:

Finally, we cannot resist calling attention to the remark of Frenkel, Lepowsky and Meurman, that vertex operator algebras (or conformal field theories) are to lattices as lattices are to codes.

I would like to understand better what the precise analogy is that is being made here.

Through my attempts to read Frenkel, Lepowsky and Meurman's book, I am aware of the story about how the "exceptional" objects,

Golay code ---> Leech lattice ---> Moonshine module,

form a hierarchy with increasingly large symmetry groups,

Mathieu group M₂₄ ---> Conway group Co₁ ---> Monster group,

and how this hierarchy led to the conjecture that uniqueness results for the Golay code and Leech lattice carry over to a uniqueness property of the Moonshine module. Frenkel, Lepowsky and Meurman speak of many analogies between the theories of codes, lattices, and vertex operator algebras. I have some understanding of the connections between codes and lattices, but so far very little understanding of vertex operator algebras and of their connection with lattices (despite having a bit of relevant physics background in conformal field theory).

My questions are

1. Are the parallels alluded to above a peculiar feature of these exceptional structures, or something that holds more generally?

2. Is there a "baby example" one can look at of these correspondences - something based on smaller and more elementary objects?