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Group Presentation for Cyclic Group Cₘₕ = Cₘ × C₂
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@Fe-r-oz
Fe-r-oz committed Oct 24, 2024
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1 change: 1 addition & 0 deletions test/Project.toml
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Expand Up @@ -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"
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213 changes: 213 additions & 0 deletions test/test_ecc_2bga_cyclicgroups_presentation.jl
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@testitem "ECC 2BGA Table 2 via Presentation of Cyclic Groups" begin
using Nemo: FqFieldElem
using Hecke: group_algebra, GF, abelian_group, gens, quo, one, GroupAlgebra
using QuantumClifford.ECC
using QuantumClifford.ECC: code_k, code_n, two_block_group_algebra_codes
using Oscar: free_group, small_group_identification, describe, order, FPGroupElem, FPGroup, FPGroupElem

function get_code(a_elts::Vector{FPGroupElem}, b_elts::Vector{FPGroupElem}, F2G::GroupAlgebra{FqFieldElem, FPGroup, FPGroupElem})
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)
return c
end

@testset "Reproduce Table 2 lin2024quantum" begin
# codes taken from Appendix C, Table 2 of [lin2024quantum](@cite)

# [[16, 2, 4]]
m = 4
F = free_group(["x", "s"])
x, s = gens(F)
G, = quo(F, [x^m, s^2, x * s * x^-1 * s^-1])
F2G = group_algebra(GF(2), G)
x, s = gens(G)
a_elts = [one(G), x]
b_elts = [one(G), x, s, x^2, s*x, s*x^3]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 16 && code_k(c) == 2
# Oscar.describe(Oscar.small_group(2*m, 2)) is C₄ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 2)

# [[16, 4, 4]]
b_elts = [one(G), x, s, x^2, s*x, x^3]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 16 && code_k(c) == 4
# Oscar.describe(Oscar.small_group(2*m, 2)) is C₄ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 2)

# [[16, 8, 2]]
a_elts = [one(G), s]
b_elts = [one(G), x, s, x^2, s*x, x^2]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 16 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(2*m, 2)) is C₄ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 2)

# [[24, 4, 5]]
m = 6
F = free_group(["x", "s"])
x, s = gens(F)
G, = quo(F, [x^m, s^2, x * s * x^-1 * s^-1])
F2G = group_algebra(GF(2), G)
x, s = gens(G)
a_elts = [one(G), x]
b_elts = [one(G), x^3, s, x^4, x^2, s*x]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 24 && code_k(c) == 4
# Oscar.describe(Oscar.small_group(2*m, 5)) is C₆ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 5)

# [[24, 12, 2]]
a_elts = [one(G), x^3]
b_elts = [one(G), x^3, s, x^4, s * x^3, x]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 24 && code_k(c) == 12
# Oscar.describe(Oscar.small_group(2*m, 5)) is C₆ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 5)

# [[32, 8, 4]]
m = 8
F = free_group(["x", "s"])
x, s = gens(F)
G, = quo(F, [x^m, s^2, x * s * x^-1 * s^-1])
F2G = group_algebra(GF(2), G)
x, s = gens(G)
a_elts = [one(G), x^6]
b_elts = [one(G), s * x^7, s * x^4, x^6, s * x^5, s * x^2]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 32 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(2*m, 5)) is C₈ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 5)

# [[32, 16, 2]]
a_elts = [one(G), s * x^4]
b_elts = [one(G), s * x^7, s * x^4, x^6, x^3, s * x^2]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 32 && code_k(c) == 16
# Oscar.describe(Oscar.small_group(2*m, 5)) is C₈ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 5)

# [[40, 4, 8]]
m = 10
F = free_group(["x", "s"])
x, s = gens(F)
G, = quo(F, [x^m, s^2, x * s * x^-1 * s^-1])
F2G = group_algebra(GF(2), G)
x, s = gens(G)
a_elts = [one(G), x]
b_elts = [one(G), x^5, x^6, s * x^6, x^7, s * x^3]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 40 && code_k(c) == 4
# Oscar.describe(Oscar.small_group(2*m, 5)) is C₁₀ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 5)

# [[40, 8, 5]]
a_elts = [one(G), x^6]
b_elts = [one(G), x^5, s, x^6 , x, s * x^2]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 40 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(2*m, 5)) is C₁₀ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 5)

# [[40, 20, 2]]
a_elts = [one(G), x^5]
b_elts = [one(G), x^5, s, x^6, s * x^5, x]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 40 && code_k(c) == 20
# Oscar.describe(Oscar.small_group(2*m, 5)) is C₁₀ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 5)

# [[48, 8, 6]]
m = 12
F = free_group(["x", "s"])
x, s = gens(F)
G, = quo(F, [x^m, s^2, x * s * x^-1 * s^-1])
F2G = group_algebra(GF(2), G)
x, s = gens(G)
a_elts = [one(G), s * x^10]
b_elts = [one(G), x^3, s * x^6, x^4, x^7, x^8]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 48 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(2*m, 9)) is C₁₂ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 9)

# [[48, 12, 4]]
a_elts = [one(G), x^3]
b_elts = [one(G), x^3, s * x^6, x^4, s * x^9, x^7]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 48 && code_k(c) == 12
# Oscar.describe(Oscar.small_group(2*m, 9)) is C₁₂ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 9)

# [[48, 16, 3]]
a_elts = [one(G), x^4]
b_elts = [one(G), x^3, s * x^6, x^4, x^7, s * x^10]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 48 && code_k(c) == 16
# Oscar.describe(Oscar.small_group(2*m, 9)) is C₁₂ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 9)

# [[48, 24, 2]]
a_elts = [one(G), s * x^6]
b_elts = [one(G), x^3, s * x^6, x^4, s * x^9, s * x^10]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C12 x C2"
@test code_n(c) == 48 && code_k(c) == 24
# Oscar.describe(Oscar.small_group(2*m, 9)) is C₁₂ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 9)

# [[56, 8, 7]]
m = 14
F = free_group(["x", "s"])
x, s = gens(F)
G, = quo(F, [x^m, s^2, x * s * x^-1 * s^-1])
F2G = group_algebra(GF(2), G)
x, s = gens(G)
a_elts = [one(G), x^8]
b_elts = [one(G), x^7, s, x^8, x^9, s * x^4]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 56 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(2*m, 4)) is C₁₄ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 4)

# [[56, 28, 2]]
a_elts = [one(G), x^7]
b_elts = [one(G), x^7, s, x^8, s * x^7, x]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*m
@test describe(G) == "C$m x C2"
@test code_n(c) == 56 && code_k(c) == 28
# Oscar.describe(Oscar.small_group(2*m, 4)) is C₁₄ x C₂, cross-check it with G
@test small_group_identification(G) == (order(G), 4)
end
end

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