[[8rp, (8r − 2)p − 2m, 4]] Delfosse-Reichardt code

[Fe-r-oz]

The paper Short Shor-style syndrome sequences has many interesting generalized quantum codes from base that could be implemented. This PR implements the second code: [[8p, 6(p-1), 4]] Delfosse Generalized [8,4,4] code. This is one of the generalized codes that were developed by Delfosse and Reichardt. The first quantum stabilizer code is implemented in #465

Interestingly, ECC Zoo don't have entries for these 'generalized codes' yet. These codes are tested using the test_ecc_base.jl for properties checkup. These codes were defined as [[8p, 6(p-1), 4]] codes so I named them DelfosseGeneralized844. Internally, it uses extended blocks of [8,4,4] Reed Muller code.

Please find the following schematics for these codes:

What's left is to add some explicit tests from the paper.

• 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.
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[Fe-r-oz]

Delfosse and Reichardt uses:

• [8, 4, 4] Reed-Muller code to create [[8p, 6(p−1), 4 ]] self-dual CSS quantum codes, for p ≥ 2
• [16,11, 4] Reed-Muller code to create [[16p, 14p − 8, 4]] self-dual CSS quantum codes, for p ≥ 1

Currently(105877a), I have implemented the Delfosse Generalized [8,4,4] code as [[8p, 6(p-1), 4]] quantum stabilizer code. The next step is to create the second bullet-point code.

To enhance the flexibility of the code, a better approach would be to generalize these codes using the Reed-Muller code as the base matrix. Instead of hardcoding the [8,4,4] or [16,11, 4] Reed-Muller codes as base matrices, we could allow users to input parameters r and m, making the code more generalized.

So, we can invent :) the following code: [[8rp, (8r−2)p−2m, 4]] Delfosse Generalized ReedMuller code

• [8, 4, 4] Reed-Muller code to create[[8p, 6(p−1), 4 ]]self-dual CSS quantum codes, for p ≥ 2
  [8,4,4] base code, r =1, m =3

julia> parity_checks(ReedMuller(1,3))
4×8 Matrix{Bool}:
 1  1  1  1  1  1  1  1
 1  1  1  1  0  0  0  0
 1  1  0  0  1  1  0  0
 1  0  1  0  1  0  1  0

[[8×1×p = 8p, (8-2)×p - 2×3 = 6p - 6 = 6(p-1), 4]] quantum stabilizer code which is the same as [[8p, 6(p−1), 4 ]]

• [16,11, 4] Reed-Muller code to create [[16p, 14p − 8, 4]] self-dual CSS quantum codes, for p ≥ 1

[16, 11,4] base code, r =2, m =4

julia> parity_checks(ReedMuller(2,4))
5×16 Matrix{Bool}:
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  0  0  0  0  0  0  0  0
 1  1  1  1  0  0  0  0  1  1  1  1  0  0  0  0
 1  1  0  0  1  1  0  0  1  1  0  0  1  1  0  0
 1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0

[[8×2p = 16p, (16-2)×p - 2×4 = 14p - 8, 4]] quantum stabilizer code which is the same as [[16p, 14p − 8, 4]]

[Fe-r-oz]

Please find the schematic for [[8rp, (8r − 2)p − 2m, 4]] Delfosse Generalized Reed-Muller code

codes = [DelfosseGeneralizedReedMuller(2, 1, 3),
         DelfosseGeneralizedReedMuller(4, 1, 3),
         DelfosseGeneralizedReedMuller(6, 1, 3),
         DelfosseGeneralizedReedMuller(2, 2, 4),
         DelfosseGeneralizedReedMuller(4, 2, 4),
         DelfosseGeneralizedReedMuller(6, 2, 4),]

[codecov]

Codecov Report

Attention: Patch coverage is 75.67568% with 9 lines in your changes missing coverage. Please review.

Project coverage is 83.00%. Comparing base (0d13791) to head (4b49e50).

Files with missing lines	Patch %	Lines
src/ecc/codes/delfosse_reichardt_code.jl	75.67%	9 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #466      +/-   ##
==========================================
+ Coverage   82.73%   83.00%   +0.26%     
==========================================
  Files          70       73       +3     
  Lines        4656     4700      +44     
==========================================
+ Hits         3852     3901      +49     
+ Misses        804      799       -5     

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[Fe-r-oz]

@Krastanov, Please help review this PR, thank you so much :)

There is a Julia error with allocated as it throws false negatives. So, I have commented out allocated that was throwing errors to avoid CI failures. 🙏🏼 Please find the detail about this Julia CI error here: #467

Edit:

P.S. Delfosse suggested to change the name to something else (changed to DelfosseReichardt to reflect the co-author as well) because we already have Generalized Reed-Muller code (this code family uses fancy operations) with 4b49e50 🙏🏼 In addition, Improved documentation and fixed typos with 2e008ae 🙏🏼