Timeline for Even lattices and binary codes
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2011 at 17:31 | comment | added | Marcel Bischoff | Yes I meant which contain $(2\mathbb Z)^n$.\ | |
| Aug 29, 2011 at 17:37 | comment | added | Noam D. Elkies | An even lattice containing $(2{\bf Z})^n$ is automatically contained in $(\frac12{\bf Z})^n$. More generally, if $L$ is an even lattice containing some lattice $L_0$ with finite index then $L \subseteq L_0^*$. This is because any even lattice is integral thanks to the usual "polarization" trick: $ \langle v,w \rangle = \frac12(\|v+w\|^2 - \|v\|^2 - \|w\|^2). $ Yes, there are even unimodular $L$ not containing $(2{\bf Z})^n$ [you must mean this, not "sublattice of $(2{\bf Z})^n$" which can never happen]; indeed for large enough $n\equiv0\bmod 8$ there need be no $v \in L$ of length $\sqrt4$. | |
| May 6, 2011 at 19:57 | comment | added | Marcel Bischoff | Yeah I know that not all even lattices are sublattices of $(2\mathbb{Z})^n$ for example $\sqrt{2}\ZZ$. I am wondering if this is also true for unimodular lattices. | |
| May 6, 2011 at 11:37 | comment | added | Roland Bacher | Even lattices do not necessarily have $(2\mathbb Z)^n$ as a sublattice. The correct definition is: An integral lattice is even if all its vectors have even norm (where the norm is the square of the usual Euclidean norm). Example $\sqrt 2\mathbb Z$ is an even lattice (which is not unimodular, even unimodular lattice exist if and only if the dimension is divisible by $8$). | |
| May 6, 2011 at 11:06 | answer | added | S. Carnahan♦ | timeline score: 5 | |
| May 6, 2011 at 9:59 | comment | added | Marcel Bischoff | Thank you. I will check this book. $(2\mathbb Z)^n$ it self is what i considered as a trivial examples. So I should maybe say I want sublattices of $(1/2\mathbb{Z})^n$ which are a proper sublattices of $(2\mathbb Z)^n$. So I have to choose a subsets of $(1/2\mathbb Z)^n / (2\mathbb Z)^n \cong \mathbb Z_4^n$ which I add, and eveness gives restrictions. | |
| May 6, 2011 at 0:44 | comment | added | Henry Cohn | I assume you want your lattices to be unimodular (which gives the restriction to dimensions that are multiples of $8$); otherwise, $(2\mathbb{Z})^n$ is itself an example in any dimension. Chapters 5 and 7 of Sphere Packings, Lattices and Groups (by Conway and Sloane) give a lot of information on how to construct lattices from codes. | |
| May 5, 2011 at 22:46 | history | edited | Steve Huntsman |
added coding theory tag
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| May 5, 2011 at 21:27 | comment | added | André Henriques | You might be interested in reading this: mathoverflow.net/questions/17617/… | |
| May 5, 2011 at 20:12 | history | asked | Marcel Bischoff | CC BY-SA 3.0 |