Group Presentation via Oscar Free Group

ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl

@@ -5,7 +5,7 @@ using DocStringExtensions
 import QuantumClifford, LinearAlgebra
 import Hecke: Group, GroupElem, AdditiveGroup, AdditiveGroupElem,
     GroupAlgebra, GroupAlgebraElem, FqFieldElem, representation_matrix, dim, base_ring,
-    multiplication_table, coefficients, abelian_group, group_algebra, small_group, direct_product
+    multiplication_table, coefficients, abelian_group, group_algebra
 import Nemo
 import Nemo: characteristic, matrix_repr, GF, ZZ, lift
 

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -161,48 +161,6 @@ function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
     LPCode(A, B)
 end
 
-"""
-[[72, 8, 9]] 2BGA code from Table 1 of [lin2024quantum](@cite) with direct product
-of `C₉ x C₄`.
-
-```jldoctest
-julia> c = two_block_group_algebra_codes([0, 28], [0, 9, 18, 12, 29, 14], (36, 2));
-
-julia> code_n(c), code_k(c)
-(72, 8)
-```
-"""
-function two_block_group_algebra_codes(a_shifts::Array{Int}, b_shifts::Array{Int}, sg::Tuple{Int,Int})
-    m, i = sg
-    g1 = small_group(m, i)
-    GA = group_algebra(GF(2), g1)
-    a = sum(GA[n%dim(GA)+1] for n in a_shifts)
-    b = sum(GA[n%dim(GA)+1] for n in b_shifts)
-    two_block_group_algebra_codes(a, b)
-end
-
-"""
-[[48, 8, 6]] 2BGA code from Table 3 of [lin2024quantum](@cite) with dihedral group of
-order `l = 12`.
-
-```jldoctest
-julia> c = two_block_group_algebra_codes([0, 10], [0, 8, 9, 4, 2, 5], (12, 4), (2, 1));
-
-julia> code_n(c), code_k(c)
-(48, 8)
-```
-"""
-function two_block_group_algebra_codes(a_shifts::Array{Int}, b_shifts::Array{Int}, sg1::Tuple{Int,Int}, sg2::Tuple{Int,Int})
-    m, i = sg1
-    l, i = sg2
-    g1 = small_group(m, i)
-    g2 = small_group(l, i)
-    GA = group_algebra(GF(2), direct_product(g1, g2))
-    a = sum(GA[n%dim(GA)+1] for n in a_shifts)
-    b = sum(GA[n%dim(GA)+1] for n in b_shifts)
-    two_block_group_algebra_codes(a, b)
-end
-
 """
 Generalized bicycle codes, which are a special case of 2GBA codes (and therefore of lifted product codes).
 Here the group is chosen as the cyclic group of order `l`,

test/Project.toml

@@ -16,6 +16,7 @@ LDPCDecoders = "3c486d74-64b9-4c60-8b1a-13a564e77efb"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 MacroTools = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
+Oscar = "f1435218-dba5-11e9-1e4d-f1a5fab5fc13"
 PyQDecoders = "17f5de1a-9b79-4409-a58d-4d45812840f7"
 Quantikz = "b0d11df0-eea3-4d79-b4a5-421488cbf74b"
 QuantumInterface = "5717a53b-5d69-4fa3-b976-0bf2f97ca1e5"

test/test_ecc_nonabelin2bga.jl

@@ -0,0 +1,253 @@
+@testitem "ECC 2BGA lin2024quantum" begin
+    import Hecke: group_algebra, GF, abelian_group, gens, quo, one
+    using QuantumClifford.ECC: LPCode, code_k, code_n, two_block_group_algebra_codes
+    using Oscar: free_group, small_group_identification, describe
+
+    @testset "Reproduce Table 1 Block 1" begin
+        # [[72, 8, 9]]
+        F = free_group(["r"])
+        r = gens(F)[1]
+        G, = quo(F, [r^36])
+        F2G = group_algebra(GF(2), G)
+        r = gens(G)[1]
+        a_elts = [one(G), r^28]
+        b_elts = [one(G), r, r^18, r^12, r^29, r^14]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C36"
+        @test small_group_identification(G) == (36, 2)
+        @test code_n(c) == 72 && code_k(c) == 8
+    end
+
+    @testset "Reproduce Table 1 Block 2" begin
+        # [[72, 8, 9]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^4, r^9, s^(-1)*r*s*r])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), r]
+        b_elts = [one(G), s, r^6, s^3 * r, s * r^7, s^3 * r^5]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C9 : C4"
+        @test small_group_identification(G) == (36, 1)
+        @test code_n(c) == 72 && code_k(c) == 8
+
+        # [[80, 8, 10]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^5, r^8, r^(-1)*s*r*s])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), s*r^4]
+        b_elts = [one(G), r, r^2, s, s^3 * r, s^2 * r^6]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C5 : C8"
+        @test small_group_identification(G) == (40, 1)
+        @test code_n(c) == 80 && code_k(c) == 8
+
+        # [[96, 8, 12]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^6, r^8, (r*s)^8])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), s*r^2]
+        b_elts = [one(G), r, s^3, s^4, s^2 * r^5, s^4 * r^6]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C5 : C8"
+        @test small_group_identification(G) == (40, 1)
+        @test code_n(c) == 80 && code_k(c) == 8
+    end
+
+    @testset "Reproduce Table 1 Block 3" begin
+        # [[54, 6, 9]]
+        F = free_group(["r"])
+        r = gens(F)[1]
+        G, = quo(F, [r^27])
+        F2G = group_algebra(GF(2), G)
+        r = gens(G)[1]
+        a_elts = [one(G), r, r^3, r^7]
+        b_elts = [one(G), r, r^12, r^19]
+        a = sum(F2G(x) for x in a_elts);
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C27"
+        @test small_group_identification(G) == (27, 1)
+        @test code_n(c) == 54 && code_k(c) == 6
+
+        # [[60, 6, 10]]
+        F = free_group(["r"])
+        r = gens(F)[1]
+        G, = quo(F, [r^30])
+        F2G = group_algebra(GF(2), G)
+        r = gens(G)[1]
+        a_elts = [one(G), r^10, r^6, r^13]
+        b_elts = [one(G), r^25, r^16, r^12]
+        a = sum(F2G(x) for x in a_elts);
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C30"
+        @test small_group_identification(G) == (30, 4)
+        @test code_n(c) == 60 && code_k(c) == 6
+
+        # [[70, 8, 10]]
+        F = free_group(["r"])
+        r = gens(F)[1]
+        G, = quo(F, [r^35])
+        F2G = group_algebra(GF(2), G)
+        r = gens(G)[1]
+        a_elts = [one(G), r^15, r^16, r^18]
+        b_elts = [one(G), r, r^24, r^27]
+        a = sum(F2G(x) for x in a_elts);
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C35"
+        @test small_group_identification(G) == (35, 1)
+        @test code_n(c) == 70 && code_k(c) == 8
+
+        # [[72, 8, 10]]
+        F = free_group(["r"])
+        r = gens(F)[1]
+        G, = quo(F, [r^36])
+        F2G = group_algebra(GF(2), G)
+        r = gens(G)[1]
+        a_elts = [one(G), r^9, r^28, r^31]
+        b_elts = [one(G), r, r^21, r^34]
+        a = sum(F2G(x) for x in a_elts);
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C36"
+        @test small_group_identification(G) == (36, 2)
+        @test code_n(c) == 72 && code_k(c) == 8
+
+        # [[72, 10, 9]]
+        F = free_group(["r"])
+        r = gens(F)[1]
+        G, = quo(F, [r^36])
+        F2G = group_algebra(GF(2), G)
+        r = gens(G)[1]
+        a_elts = [one(G), r^9, r^28, r^13]
+        b_elts = [one(G), r, r^3, r^22]
+        a = sum(F2G(x) for x in a_elts);
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C36"
+        @test small_group_identification(G) == (36, 2)
+        @test code_n(c) == 72 && code_k(c) == 10
+    end
+
+    @testset "Reproduce Table 1 Block 4" begin
+        # [[72, 8, 9]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^4, r^9, s^(-1)*r*s*r])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), s, r, s*r^6]
+        b_elts = [one(G), s^2*r, s^2*r^6, r^2]
+        a = sum(F2G(x) for x in a_elts);
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C9 : C4"
+        @test small_group_identification(G) == (36, 1)
+        @test code_n(c) == 72 && code_k(c) == 8
+
+        # [[80, 8, 10]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^5, r^8, r^(-1)*s*r*s])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), r, s, s^3*r^5]
+        b_elts = [one(G), r^2, s*r^4, s^3*r^2]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C5 : C8"
+        @test small_group_identification(G) == (40, 1)
+        @test code_n(c) == 80 && code_k(c) == 8
+
+        # [[96, 8, 12]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^3, r^16, r^(-1)*s*r*s])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), r, s, r^14]
+        b_elts = [one(G), r^2, s*r^4, r^11]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C3 : C16"
+        @test small_group_identification(G) == (48, 1)
+        @test code_n(c) == 96 && code_k(c) == 6
+
+        # [[80, 9, 9]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^4, r^10, (r*s)^2])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), r^7, r^8, s*r^10]
+        b_elts = [one(G), s, r^5, s^2*r^13]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C7 x S3"
+        @test small_group_identification(G) == (42, 3)
+        @test code_n(c) == 84 && code_k(c) == 10
+
+        # [[84, 10, 9]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^3, r^14, r^(-1)*s*r*s])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), r^7, r^8, s*r^10]
+        b_elts = [one(G), s, r^5, s^2*r^13]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C7 x S3"
+        @test small_group_identification(G) == (42, 3)
+        @test code_n(c) == 84 && code_k(c) == 10
+
+        # [[96, 6, 12]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^4, r^12, s^(-1)*r*s*r])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), s, r^9, s * r]
+        b_elts = [one(G), s^2 * s^9, r^7, r^2]
+        a = sum(F2G(x) for x in a_elts);
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C12 : C4"
+        @test small_group_identification(G) == (48, 13)
+        @test code_n(c) == 96 && code_k(c) == 6
+
+        # [[96, 12, 10]]
+        F = free_group(["r", "s"])
+        r, s = gens(F)
+        G, = quo(F, [s^6, r^8, r^(-1)*s*r*s])
+        F2G = group_algebra(GF(2), G)
+        r, s = gens(G)
+        a_elts = [one(G), r, s^3 * r^2, s^2 * r^3]
+        b_elts = [one(G), r, s^4 * r^6, s^5 * r^3]
+        a = sum(F2G(x) for x in a_elts)
+        b = sum(F2G(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a, b)
+        @test describe(G) == "C2 x (C3 : C8)"
+        @test small_group_identification(G) == (48, 9)
+        @test code_n(c) == 96 && code_k(c) == 12
+    end
+end