Coprime Bivariate Bicycle code via Hecke's Group Algebra (#378)

Co-authored-by: Stefan Krastanov <[email protected]>

Author: Fe-r-oz

docs/src/references.bib

@@ -524,6 +524,13 @@ @article{lin2024quantum
   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}
+}
+
 @misc{voss2024multivariatebicyclecodes,
   title={Multivariate Bicycle Codes}, 
   author={Lukas Voss and Sim Jian Xian and Tobias Haug and Kishor Bharti},

docs/src/references.md

@@ -40,6 +40,7 @@ For quantum code construction routines:
 - [steane1999quantum](@cite)
 - [campbell2012magic](@cite)
 - [anderson2014fault](@cite)
+- [wang2024coprime](@cite)
 - [voss2024multivariatebicyclecodes](@cite)
 - [lin2024quantum](@cite)
 - [bravyi2024high](@cite)

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -207,6 +207,7 @@ julia> c = two_block_group_algebra_codes(A,B);
 julia> code_n(c), code_k(c)
 (756, 16)
 ```
+
 ### Multivariate Bicycle code
 
 The group algebra of the qubit multivariate bicycle (MB) code with r variables is `𝔽₂[𝐺ᵣ]`,
@@ -231,6 +232,32 @@ julia> code_n(c), code_k(c)
 (48, 4)
 ```
 
+### Coprime Bivariate Bicycle code
+
+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 generator `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`.
+
+[108, 12, 6]] coprime-bivariate bicycle (BB) code from Table 2 of [wang2024coprime](@cite).
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens;
+
+julia> l=2; m=27;
+
+julia> GA = group_algebra(GF(2), abelian_group([l*m]));
+
+julia> 𝜋 = gens(GA)[1];
+
+julia> A = 𝜋^2 + 𝜋^5  + 𝜋^44;
+
+julia> B = 𝜋^8 + 𝜋^14 + 𝜋^47;
+
+
+(108, 12)
+```
+
 See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref).
 """
 function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)

test/test_ecc_base.jl

@@ -58,6 +58,25 @@ A[LinearAlgebra.diagind(A, 5)] .= GA(1)
 B = reshape([1 + x + x^6], (1, 1))
 push!(other_lifted_product_codes, LPCode(A, B))
 
+# coprime Bivariate Bicycle codes from Table 2 of [wang2024coprime](@cite)
+# [[108,12,6]]
+l=2; m=27
+GA = group_algebra(GF(2), abelian_group([l*m]))
+𝜋 = gens(GA)[1]
+A = 𝜋^2 + 𝜋^5  + 𝜋^44
+B = 𝜋^8 + 𝜋^14 + 𝜋^47
+coprimeBB1 = two_block_group_algebra_codes(A, B)
+
+# [[126,12,10]]
+l=7; m=9
+GA = group_algebra(GF(2), abelian_group([l*m]))
+𝜋 = gens(GA)[1]
+A = 1   + 𝜋    + 𝜋^58
+B = 𝜋^3 + 𝜋^16 + 𝜋^44
+coprimeBB2 = two_block_group_algebra_codes(A, B)
+
+test_coprimeBB_codes = [coprimeBB1, coprimeBB2]
+
 # Multivariate Bicycle codes taken from Table 1 of [voss2024multivariatebicyclecodes](@cite)
 # Weight-4 [[144, 2, 12]] MBB code
 l=8; m=9
@@ -131,7 +150,7 @@ const code_instance_args = Dict(
     :CSS => (c -> (parity_checks_x(c), parity_checks_z(c))).([Shor9(), Steane7(), Toric(4, 4)]),
     :Concat => [(Perfect5(), Perfect5()), (Perfect5(), Steane7()), (Steane7(), Cleve8()), (Toric(2, 2), Shor9())],
     :CircuitCode => random_circuit_code_args,
-    :LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, test_bb_codes, test_mbb_codes, other_lifted_product_codes)),
+    :LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, test_bb_codes, test_mbb_codes, test_coprimeBB_codes, other_lifted_product_codes)),
     :QuantumReedMuller => [3, 4, 5]
 )
 

test/test_ecc_coprime_bivaraite_bicycle.jl

@@ -0,0 +1,59 @@
+@testitem "ECC coprime Bivaraite Bicycle" begin
+    using Nemo
+    using Nemo: gcd
+    using Hecke
+    using Hecke: group_algebra, GF, abelian_group, gens
+    using QuantumClifford.ECC: two_block_group_algebra_codes, code_k, code_n
+
+    @testset "Reproduce Table 2 wang2024coprime" begin
+        # [[30,4,6]]
+        l=3; m=5;
+        GA = group_algebra(GF(2), abelian_group([l*m]))
+        𝜋 = gens(GA)[1]
+        A = 1 + 𝜋   + 𝜋^2
+        B = 𝜋 + 𝜋^3 + 𝜋^8
+        c = two_block_group_algebra_codes(A, B)
+        @test gcd([l,m]) == 1
+        @test code_n(c) == 30 && code_k(c) == 4
+
+        # [[42,6,6]]
+        l=3; m=7;
+        GA = group_algebra(GF(2), abelian_group([l*m]))
+        𝜋 = gens(GA)[1]
+        A = 1 + 𝜋^2 + 𝜋^3
+        B = 𝜋 + 𝜋^3 + 𝜋^11
+        c = two_block_group_algebra_codes(A, B)
+        @test gcd([l,m]) == 1
+        @test code_n(c) == 42 && code_k(c) == 6
+
+        # [[70,6,8]]
+        l=5; m=7;
+        GA = group_algebra(GF(2), abelian_group([l*m]))
+        𝜋 = gens(GA)[1]
+        A = 1 + 𝜋 + 𝜋^5;
+        B = 1 + 𝜋 + 𝜋^12;
+        c = two_block_group_algebra_codes(A, B)
+        @test gcd([l,m]) == 1
+        @test code_n(c) == 70 && code_k(c) == 6
+
+        # [[108,12,6]]
+        l=2; m=27;
+        GA = group_algebra(GF(2), abelian_group([l*m]))
+        𝜋 = gens(GA)[1]
+        A = 𝜋^2 + 𝜋^5  + 𝜋^44
+        B = 𝜋^8 + 𝜋^14 + 𝜋^47
+        c = two_block_group_algebra_codes(A, B)
+        @test gcd([l,m]) == 1
+        @test code_n(c) == 108 && code_k(c) == 12
+
+        # [[126,12,10]]
+        l=7; m=9
+        GA = group_algebra(GF(2), abelian_group([l*m]))
+        𝜋 = gens(GA)[1]
+        A = 1   + 𝜋    + 𝜋^58
+        B = 𝜋^3 + 𝜋^16 + 𝜋^44
+        c = two_block_group_algebra_codes(A, B)
+        @test gcd([l,m]) == 1
+        @test code_n(c) == 126 && code_k(c) == 12
+    end
+end