Multivariate Multicycle codes using Koszul Complexes

[Fe-r-oz]

We introduce a novel class of quantum CSS codes — Multivariate Multicycle codes — constructed from multivariate polynomial quotient ring formalism over finite fields over GF(2). Our discovery establishes that the boundary maps of these codes are governed by the combinatorial structure of Koszul complexes. Specifically, for a code defined by t polynomial relations, we demonstrate that the k-th boundary map is obtained by taking the Koszul matrix in degree k and replacing each variable entry with the corresponding circulant matrix.

This family of codes generalizes the following codes and it enables full single shot decoding in both X and Z directions and resolves #603

1. Bivariate Bicycle Codes
2. Trivariate Tricycle Codes
3. Tetravariate Tetracycle codes

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[codecov]

Codecov Report

❌ Patch coverage is 85.39326% with 13 lines in your changes missing coverage. Please review.
✅ Project coverage is 76.67%. Comparing base (73c133a) to head (fc86310).

Files with missing lines	Patch %	Lines
...QuantumCliffordOscarExt/multivariate_multicycle.jl	85.71%	12 Missing ⚠️
src/ecc/codes/qeccs_using_oscar.jl	80.00%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #606      +/-   ##
==========================================
+ Coverage   76.53%   76.67%   +0.14%     
==========================================
  Files         122      123       +1     
  Lines        7491     7579      +88     
==========================================
+ Hits         5733     5811      +78     
- Misses       1758     1768      +10     

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