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Lattices over a finite field?
In https://en.wikipedia.org/wiki/Lattice-based_cryptography it says that Lattice Cryptography uses lattices over a finite field, but in the definition of a lattice here, lattice are defined over an infinite field R^n. Can lattices be defined over a finite field instead of over R^n? — Preceding unsigned comment added by 22.214.171.124 (talk) 11:07, 11 September 2015 (UTC)
- In the "First Examples" section of the Lattice (discrete subgroup) article, it talks about the the parent group being discrete. It may be that the lattices used in lattice cryptography fit better under the definition in that article.
- Tangentially, this article and the Lattice (discrete subgroup) article seem to overlap quite a bit to my non-mathematician eye. Sanpitch (talk) 16:20, 25 June 2019 (UTC)
- This reference to "lattices over finite fields" (whatever that may refer to) has disappeared from the article. I don't think it made sense originally. Regarding the remark about this article overlapping with Lattice (discrete subgroup): this is the case, as the lattices discussed in this article are a special case of the general notion of a lattice in a topological group, the latter being in this case. There are different flavours to the general theory and the euclidean one is perhaps the best-known and the simplest, and the one with the most immediate applications, so in my opinion it is important to have a separate article for it (I'm not a fan of the title however, maybe "Euclidean lattice" would be a better fit but that is not very important). jraimbau (talk) 10:42, 26 June 2019 (UTC)
I'm getting really frustrated trying to understand the contents of this wikipedia page. At the beginning of the page and throughout the majority of the page, a lattice in R^n is a subgroup of the additive group R^n generated by a basis of the vector space R^n. Ok, fine. And then what the hell is going on in "Lattices in two dimensions: detailed discussion" (and "Lattices in 3 dimensions")? After getting a headache, I realize that the only explanation is that a different definition of lattice is considered in these paragraphs, but it isn't written anywhere in the page, making it incomprehensible. I assume the definition considered there is something like "a discrete subgroup of the isometry group of R^n" (and not just an additive subgroup of R^n) possibly with an extra condition like: whose subgroup of translations is a lattice of R^n in the previous sense. The paragraph starts right off the bat with "There are five 2D lattice types". Oh yeah? What's a lattice type? And what is the definition of a lattice you're using all of a sudden? By the way, if it's a wallpaper group, then according to the corresponding wikipedia page there are 17 wallpaper groups, not 5, so what's up with that? — Preceding unsigned comment added by Seub (talk • contribs) 20:27, 11 November 2018 (UTC)
- Looking briefly at these sections it seems that they classify 2- and 3-dimensional lattices according to their group of Euclidean symmetries. jraimbau (talk) 14:02, 13 November 2018 (UTC)
Lattices in complex space
"For example, the Gaussian integers form a lattice in C^n"
- I don't claim to be a number theorist, yet I definitely do think that "the Gaussian integers form a lattice in ". One nice basis for this lattice is the identity matrix :-)
- I'm less convinced about the next statement "every lattice in is a free abelian group of rank 2n." I guess the rank should be n. Sanpitch (talk) 19:23, 26 December 2019 (UTC)
Source for definition in 'Lattices in general vector-spaces' section
While the definition and properties given in this section seem reasonable, there's no source given for the definition. Is the restriction to the finite case generally accepted? While I see no applications for the infinite case, the definitions and properties given here could straightforwardly be generalised to, say, the sequence space l2, with a caveat about the dimension theorem for vector spaces, which makes me wonder about the canonicity of the definition. — Charles Stewart (talk) 07:49, 27 May 2021 (UTC)
- This does not have seem to be studied, at least I'm unaware of any work on ZZ-submodules of infinite-dimensional vector spaces and searching "lattices in Hilbert space" or "lattices in infinite-dimensional vector spaces" returns unrelated results (e.g. https://doi.org/10.1017/S0305004100076295). I'm no functional analyst but this seems to indicate that the topic is extremely niche if it exists at all.
- In addition I do not see a straightforward definition of a lattice in a general topological vector space that retains the properties of finite-dimensional lattices. To take the example of infinite-dimensional Hilbert space you could either pick the ZZ-submodule generated by a Hilbert basis or that generated by a vector space basis. In the first case you do get a discrete subset but it does not have a bounded fundamental domain. In the second case you get something that is likely dense in your vector space. jraimbau (talk) 10:17, 28 May 2021 (UTC)
I too would greatly appreciate a reference for this section! (although in my case I am not interested in the infinite-dimensional case) I have added a "references needed" tag to the article (but knowing the mathematics corner of Wikipedia, I am not getting my hopes up that this will have any effect) Joel Brennan (talk) 23:05, 6 April 2022 (UTC)
Cleaning up paragraph in "Symmetry considerations and examples"
Can someone clean up the second paragraph in section "Symmetry considerations and examples"? It seems like it's trying to pack too much information in a sentence, which makes it nearly incomprehensible to me:
"A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense."