Lifted Product construction of CoPrime Bivariate Bicycle codes

QuantumSavory · Oct 4, 2024 · 82caa49 · 82caa49
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diff --git a/docs/Project.toml b/docs/Project.toml
@@ -8,6 +8,7 @@ Graphs = "86223c79-3864-5bf0-83f7-82e725a168b6"
 Hecke = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
 LDPCDecoders = "3c486d74-64b9-4c60-8b1a-13a564e77efb"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
+Oscar = "f1435218-dba5-11e9-1e4d-f1a5fab5fc13"
 Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
 PyQDecoders = "17f5de1a-9b79-4409-a58d-4d45812840f7"
 Quantikz = "b0d11df0-eea3-4d79-b4a5-421488cbf74b"

diff --git a/docs/src/references.bib b/docs/src/references.bib
@@ -487,3 +487,11 @@ @article{anderson2014fault
   year={2014},
   publisher={APS}
 }
+
+@article{wang2024coprime,
+  title={Coprime Bivariate Bicycle Codes and their Properties},
+  author={Wang, Ming and Mueller, Frank},
+  journal={arXiv preprint arXiv:2408.10001},
+  year={2024}
+}
+
diff --git a/docs/src/references.md b/docs/src/references.md
@@ -40,6 +40,7 @@ For quantum code construction routines:
 - [steane1999quantum](@cite)
 - [campbell2012magic](@cite)
 - [anderson2014fault](@cite)
+- [wang2024coprime](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

diff --git a/ext/QuantumCliffordHeckeExt/lifted_product.jl b/ext/QuantumCliffordHeckeExt/lifted_product.jl
@@ -70,6 +70,52 @@ julia> code_n(c2), code_k(c2)
 - When the base matrices of the `LPCode` are 1×1 and their elements are sums of cyclic permutations, the code is called a generalized bicycle code [`generalized_bicycle_codes`](@ref).
 - When the two matrices are adjoint to each other, the code is called a bicycle code [`bicycle_codes`](@ref).
 
+
+# Examples
+
+The coprime bivariate bicycle (BB) codes are defined by two polynomials `𝑎(𝑥,𝑦)` and `𝑏(𝑥,𝑦)`, where `𝑙` and `𝑚` are coprime, and can be expressed as univariate polynomials `𝑎(𝜋)` and `𝑏(𝜋)`, with `𝜋 = 𝑥𝑦`. They can be viewed as a special case of Lifted Product construction based on abelian group `ℤₗ x ℤₘ` where `ℤⱼ` cyclic group of order `j` with generator `𝜋`.
+
+[108, 12, 6]] coprime-bivariate bicycle (BB) codes from Table 2 of [wang2024coprime](@cite).
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens; import Oscar: SubPcGroup;
+
+julia> l=2; m=27;
+
+julia> GA = group_algebra(GF(2), abelian_group(SubPcGroup, [l*m]));
+
+julia> π = gens(GA)[1];
+
+julia> A = reshape([π^2 + π^5 + π^44], (1,1));
+
+julia> B = reshape([π^8 + π^14 + π^47], (1,1));
+
+julia> c1 = LPCode(A, B);
+
+julia> code_n(c1), code_k(c1)
+(108, 12)
+```
+
+[126, 12, 10]] coprime-bivariate bicycle (BB) codes from Table 2 of [wang2024coprime](@cite).
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens; import Oscar: SubPcGroup;
+
+julia> l=7; m=9;
+
+julia> GA = group_algebra(GF(2), abelian_group(SubPcGroup, [l*m]));
+
+julia> π = gens(GA)[1];
+
+julia> A = reshape([1 + π + π^58], (1,1));
+
+julia> B = reshape([π^3 + π^16 + π^44], (1,1));
+
+julia> c1 = LPCode(A, B);
+
+julia> code_n(c1), code_k(c1)
+(126, 12)
+```
 ## The representation function
 
 We use the default representation function `Hecke.representation_matrix` to convert a `GF(2)`-group algebra element to a binary matrix.