I know that good codes allow you to construct good sphere packings in Euclidean space, e.g. the binary Golay code is the key feature of the construction of the Leech lattice.
Going in the other direction, I am wondering whether sphere packings in $\mathbf{R}^n$ actually help with designing good codes (for some definition of "code" that is useful in a practical setting). I know that codes on $n$ bits are sphere packings in Hamming space, but I'm wondering specifically about Euclidean sphere packings.
I know that kissing configurations yield spherical codes (by definition). Are these actually useful in practice? Also, do dense (lattice) packings produce any practically useful codes? From searching on Google I found a small section of a paper of H. Cohn (page 6 of https://arxiv.org/pdf/1003.3053.pdf) which gives a simplified example of how one can do coding with noise using good sphere packings, but I am wondering whether this is actually done in practice.
Best Answer
The quick answer is no, this approach to code design in not used in practice. Take a look at the codes used in 5G or WiFi for example. Both are large scale projects where code performance and decoder complexity are key...neither use sphere packings in the code design