Timeline for What is "automorphism group of an error-correcting code" ?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2021 at 4:12 | answer | added | Symbol 1 | timeline score: 2 | |
| Mar 8, 2012 at 17:31 | vote | accept | Alexander Chervov | ||
| Mar 6, 2012 at 17:09 | comment | added | Emil Jeřábek | The @user message notification is a SE 2.0 feature which does not work on MO at all. | |
| Mar 6, 2012 at 10:55 | answer | added | Jyrki Lahtonen | timeline score: 6 | |
| Mar 5, 2012 at 7:41 | comment | added | Alexander Chervov | @Noam, Will Thank you very much for yours answers ! Just to be 100% sure - is it standard terminology ? If we consider general Hamming code of lengh 2^n is there some nice automorphism group ? PS @Noam if you kindly post comment as an answer, I will be happy to accept it. | |
| Mar 4, 2012 at 20:18 | comment | added | Noam D. Elkies | @Will: Thanks for the hint about the @ sign - and the envelope, to which I hadn't paid any attention up to now. As for code and lattice automorphism groups, already $D_4$ has non-monomial automorphisms; but when $C$ has no nonzero words of length $4$ or less, as happens for the binary Golay code, the code automorphisms together with $\lbrace \pm 1 \rbrace^n$ do account for all the lattice automorphisms. | |
| Mar 4, 2012 at 20:04 | comment | added | Will Jagy | @Noam, I suspected something might be different, so I just mentioned the books. I've never paid attention to codes...Oh, about use of the @ sign, Gerry Myerson told me on MSE that the first three characters of the other person's on-site name are what matter, as far as them being informed by the software that there is a message and where it is. I'm guessing it is similar on MO. It is hard to tell if I see a message before the usual delay about being informed (the little envelope at the top of the page, next to my name, turns orange). | |
| Mar 4, 2012 at 19:44 | comment | added | Noam D. Elkies | @W.Jagy: Aut$(C)$ is certainly contained in the isometries of the associated lattice (modulo $\lbrace \pm 1 \rbrace^n$), but sometimes there are other automorphisms: $E_7$ and $E_8$ have more isometries than you get from the Hamming and extended Hamming codes. | |
| Mar 4, 2012 at 19:31 | comment | added | Will Jagy | I think it is the integral isometry group of an associated lattice. The three books I have in this direction are SPLAG by Conway and Sloane, more introductory Lattices and Codes by Wolfgang Ebeling, also From Error Correcting Codes Through Sphere Packing to Simple Groups by Thomas M. Thompson. I quite like the Ebeling book, see my answers to my own question at mathoverflow.net/questions/69444/… | |
| Mar 4, 2012 at 19:23 | comment | added | Noam D. Elkies | For a binary code $C$ the automorphism group is just the group of permutations of the coordinates that permute $C$. (For a code of length $n$ over an arbitrary field $k$, it's the stabilizer of $C$ in the semidirect product of $S_n$ with $(k^*)^n$; when $|k|=2$ you don't see the $k^*$ factor.) For the Hamming code of length $7$ this is $GL_3({\bf Z}/2{\bf Z})$; for the extended Hamming code of length $8$, it's affine $GL_3$. This should be thoroughly explained in standard texts such as MacWilliams and Sloane. | |
| Mar 4, 2012 at 19:05 | history | edited | Alexander Chervov | CC BY-SA 3.0 |
added 10 characters in body
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| Mar 4, 2012 at 18:57 | history | asked | Alexander Chervov | CC BY-SA 3.0 |