Semi Direct Product of Groups via Oscar for 2BGA given ℤₗ⋉ℤₘ and presentation

[Fe-r-oz]

Given a group presentation, and cyclic groups of order l, m, etc, this PR introduces Semi Direct Product of Groups via Oscar for the group structure required by the code construction. This PR aims to define all the 2BGA codes that are currently not supported by the HeckeExt. Currently, the GroupAlgebra only supports one cyclic group, meaning GroupAlgebra(GF(2), abelian_group(l)) but not group structure for the following cases which are covered in this PR:

• SemiDirect Product of Group for two cyclic groups
• SemiDirect Product of Group for more than two cyclic groups
• SemiDirect and Direct Products (both) of Group for more than two cyclic groups

These cases can be seen from Table 1 of https://arxiv.org/pdf/2306.16400.

I have utilized Direct Product of Groups in previous PRs which can be similarly applied here as well. In addition, these codes rely on the use of Semi Direct Product of Groups, which is found in Oscar. As seen for example aforementioned instances, most of the codes uses structure that is given by Semi Direct Product of Groups with some Direct Product of Groups. The previous PRs introduced the use of Direct Product of Groups which can accurately use the code instance that uses Direct Product. In this PR, Semi Direct Product of Groups is covered.

I think Oscar will be needed as a weak dependency for HeckeExt for all the cases described above to be used for circuit simulation. The authors has also considered some non-abelian groups, such as Dihedral Group with this presentation for defining direct products of these groups as well. Other groups in consideration are: Alternating Group, Symmetric Group for Direct Product of groups.

They used GAP for these calculations and defining group structures in the table, so I will have to check the methods in GAP as well. Specifically, StructureDescription(SmallGroup(ℓ,#)) from GAP was used by them. The documentation for StructureDescription from GAP.

Huge thanks to Stevell from the Oscar team for valuable feedback about this generalization of direct product of groups.

• 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.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 82.99%. Comparing base (4e06c01) to head (e02bec8).
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #382      +/-   ##
==========================================
- Coverage   83.03%   82.99%   -0.05%     
==========================================
  Files          70       70              
  Lines        4404     4404              
==========================================
- Hits         3657     3655       -2     
- Misses        747      749       +2     

Flag	Coverage Δ
	?

Flags with carried forward coverage won't be shown. Click here to find out more.

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.