Issue #40513, #40514: Allow zero code and add intersection method

[Hapsa21]

This pull request resolves two issues by allowing the creation of the zero linear code and adding a method to compute the intersection of two codes.

• Fixes Allow the construction of the zero linear code #40513: The __init__ method for LinearCode is updated to correctly handle zero matrices and zero-dimensional vector space inputs, creating a valid [n, 0] code instead of raising an error.
• Fixes Intersections of linear codes #40514: A new intersection() method is added to LinearCode, implemented using the identity C1 ∩ C2 = (C1_dual + C2_dual)_dual.

All relevant doctests have been added or updated to reflect these changes.

Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

Dependencies

None.

[Hapsa21]

Looks like the automated "build documentation (PDF)" check is failing. The error log shows:

buildx failed with: ERROR: failed to build: invalid tag ".../Hapsa21/sage/...": repository name must be lowercase

It seems the build script is not converting my username to lowercase, which is causing the process to fail. This appears to be an issue with the CI workflow rather than the code in this PR.

[cxzhong]

I think the file abstract_code would also reject zero linear code

[Hapsa21]

The changes for abstract_code.py are now included, and all the local tests are passing.

[cxzhong]

LGTM

[cxzhong]

And can you test the dual code of zero linear code? @Hapsa21

[cxzhong]

And it is better for all tests. you can add the zero linear code in it

[Hapsa21]

I've added the test for the dual of the zero code and pushed the commit.

[user202729]

Use Issue instead of Trac.

(on this note, if you got the suggestion to use Trac from AI: read https://doc.sagemath.org/html/en/developer/ instead of asking AI.)

What's the reason behind the behavioral change? As far as I can see, neither of them use zero code or intersection method?

[Hapsa21]

@user202729 the changes in abstract_code.py are a side effect.

To fix the zero code bug, I had to make LinearCode always use a standard (echelon form) matrix. This caused the example outputs in the old abstract_code.py tests to change, so I updated them to match the new correct behavior.

[cxzhong]

Use Issue instead of Trac.

(on this note, if you got the suggestion to use Trac from AI: read https://doc.sagemath.org/html/en/developer/ instead of asking AI.)

What's the reason behind the behavioral change? As far as I can see, neither of them use zero code or intersection method?

I think we can construct zero linear codes from dual codes. But we can not construct a zero linear code directly. I think we can allow direct construct a zero linear code if there are no conflict with the system.

[github-actions]

Documentation preview for this PR (built with commit 7e86f9e; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[cxzhong]

some operations like decode and encode are compatible with zero linear code?

[Hapsa21]

some operations like decode and encode are compatible with zero linear code?

I've added the tests for encode and decode with the zero code. Thanks for the suggestion!

[cxzhong]

And just see this. @orlitzky Do you think we need to permit the construction of zero linear code? I think it is ok. Because we can use dual code to construct a zero linear code. But now the sage prevents our construct a zero linear code directly. If there are no conflicts. I think it can be