#6452: Codes over rings
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Reporter: wdj | Owner: rlm
Type: | Status: needs_work
enhancement | Milestone: sage-6.9
Priority: major | Resolution:
Component: | Merged in:
coding theory | Reviewers:
Keywords: | Work issues:
Authors: | Commit:
Report Upstream: N/A | 1177056cb4942b0ce938a30537357bcb07f1bf19
Branch: | Stopgaps:
public/6452 |
Dependencies: |
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Comment (by vdelecroix):
Replying to [comment:19 jsrn]:
> > True. It would basically be equivalent to implement a generalized
echelon form for matrices over `ZZ/nZZ` that give you a "pseudo-basis" of
submodules of `(ZZ/nZZ)^r`. I have no idea where to find such algorithm.
>
> It's interesting: I'll think more on this. For instance, would the
Hermite Normal form be sufficient?
I don't think that it is enough to determine the structure. An example of
a problem is if `R=ZZ/4ZZ` and a matrix of the form
{{{
2 * *
0 2 *
0 0 2
}}}
If you have a vector which is a sum of its row with coefficients `0` or
`2` then the leading coefficient vanish... and you lose the nice Hermite
form!
But I guess that the Smith normal form would be of more help. If I
intrepret correctly what I read, it tells you that any submodule of `R^r`
where `R = ZZ/nZZ` is actually of the form `R / (I1) + R / (I2) + ... + R
/ (Ik)` for some ideals `I1`, ..., `Ik`. These ideals are basically the
product `SAT` (which is diagonal) and the decomposition is explicitly
given by `T` (my source and notations from
[https://en.wikipedia.org/wiki/Smith_normal_form wikipedia]).
--
Ticket URL: <http://trac.sagemath.org/ticket/6452#comment:20>
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