Add Oscar for Free Group Functionality

[Fe-r-oz]

This aims to add Oscar as a weak dependency as Hecke, AbstractAlgebra, or Nemo does not support the following when constructing GroupAlgebra:

1. Semi Direct Product of two Groups
2. Semi Direct Product of more than Groups
3. Dihedral Groups
4. Alternating Groups
5. Direct Products of the Dihedral Group, Alternating Group, and Symmetric Group

The GroupAlgebra for the 2BGA codes require several of the aforementioned features as mentioned in #382. Even Hecke's abelian_group uses group types from Oscar, namely PcGroup, SubPcGroup, for Direct Product of Groups.

Since Oscar also provides GAP integration, this also provide the toolkit to do the following. In #382, the authors use GAP function call to determine the group structure. This can be done as follows:

julia> Oscar.GAP.Globals.StructureDescription(Oscar.GAP.Globals.SmallGroup(36,1))
GAP: "C9 : C4"
# : refers to Semi Direct Product

julia> Oscar.GAP.Globals.StructureDescription(Oscar.GAP.Globals.SmallGroup(42,3))
GAP: "C7 x S3"

julia> Oscar.GAP.Globals.StructureDescription(Oscar.GAP.Globals.SmallGroup(36,2))
GAP: "C36"

Furthermore, many paper use the GAP QDistRnd package for minimum distance calculation for QLDPC codes. Using Oscar, we can have access to this package as well. Panteleev used QDistRnd and GNU's http://www.gnu.org/software/glpk/glpk.html for minimum distance calculation.

julia> Oscar.GAP.Packages.install("https://github.com/QEC-pages/QDistRnd.git");
julia> Oscar.GAP.Packages.load("QDistRnd")
true

julia> Oscar.GAP.Globals.DistRandCSS
GAP: function( GX, GZ, num, mindist, opt... ) ... end

• The code is properly formatted and commented.
• Substantial new functionality is documented within the docs.
• All new functionality is tested.
• All of the automated tests on github pass.

[Fe-r-oz]

This is related to oscar-system/GAP.jl#960

Starting tests with 5 threads out of `Sys.CPU_THREADS = 4`...
ERROR: LoadError: InitError: GAP.jl currently does not support multithreaded garbage collection. Please run julia with `--gcthreads=1` for now.

It seems that I should use GAP.jl 0.11.2 and later instead of currently ⌅ [c863536a] GAP v0.10.4

[Fe-r-oz]

Hi, @lgoettgens, could you help with resolving the OSCAR related compatibility issues and weird CI errors? That would be greatly appreciated. Pardon for the ping.

You are closely integrated into OSCAR ecosystem and have better insights about these CI errors:

Starting tests with 5 threads out of `Sys.CPU_THREADS = 4`...
ERROR: LoadError: InitError: GAP.jl currently does not support multithreaded garbage collection. Please run julia with `--gcthreads=1` for now.
ERROR: LoadError: Windows is not supported, please try the Windows Subsystem for Linux.
For details please check: https://www.oscar-system.org/install/

Your insights about this would be greatly appreciated: #383 (comment). Thank you for your time.

[lgoettgens]

Hi @Fe-r-oz,

some comments to your above questions:

• ERROR: LoadError: Windows is not supported, please try the Windows Subsystem for Linux.: Oscar, in particular GAP.jl does not work on Windows. You need to check in this QuantumClifford's CI setup that Oscar is not used in any Windows job.
• ERROR: LoadError: InitError: GAP.jl currently does not support multithreaded garbage collection. Please run julia with --gcthreads=1 for now.: As the message already tells you, GAP.jl v0.10.x does not support multithreaded garbage collection. The message contains a setting to circumvent this on the julia side. GAP.jl v0.11.x (and v0.12.x) fixed this restriction, but you will need to wait for Oscar v1.2.0 release to get access to it.
• If you don't want to use Oscar and GAP, but first only Hecke: there is Hecke.small_group(i, j) which gives you the j-th group of order i. If you have some few fixed group constructions, you could just lookup which ID in this database correspond to the groups you are interested in. Of course, this is a hard restriction compared to using Oscar/GAP groups.
• One part of Oscar philosophy is no need to use GAP.Globals.xxx calls. Instead, Oscar provides snake_case functions for a lot of things already. E.g. Oscar.GAP.Globals.StructureDescription(Oscar.GAP.Globals.SmallGroup(36,1)) can be written as describe(small_group(36,1)) with the additional benefit that this returns a julia string (instead of a GAP string).
• Similar to the previous point, only the Oscar objects are properly typed (GAP.Globals.SymmetricGroup(3) returns a GapObj while symmetric_group(3) returns a PermGroup), and group_algebra only takes the properly typed ones.

I wanted to write another bullet point but forgot what... ah damn...

[Fe-r-oz]

Indeed. The GAP support was the side thing that was not really required.

I think the last point contained the answer to 'What about Direct Products of Groups, Semi Direct Products of Groups, as they are essential to the construction of Group Algebra shown in #382 and in many other places for our requirements? For example, Using abelian_group(PcGroup, [l,m]) is quite helpful for direct products. Maybe direct_product helps in this case. Ah... found something simpler than direct_product! But in the semi direct product case, I am not sure. Indeed, Using multiplication table is a GAP way of doing things :), and maybe it's a hard restriction for some new users.

[Fe-r-oz]

Closing in favor of #400