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tests: Specific Group Presentations Using Oscar's Free Group for Non-Abelian and Abelian Groups #390

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tests: Specific Group Presentations Using Oscar's Free Group for Non-Abelian and Abelian Groups #390

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4724811
introducing non-abelian groups via Hecke's small group with multiplic…
Fe-r-oz Oct 14, 2024
aa12524
add method to construct abelian direct product of groups via small group
Fe-r-oz Oct 14, 2024
6304dc6
Merge branch 'QuantumSavory:master' into nonabelian
Fe-r-oz Oct 18, 2024
279a175
Group Presentation via Oscar Free Group
Fe-r-oz Oct 18, 2024
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11 changes: 11 additions & 0 deletions docs/src/references.bib
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Original file line number Diff line number Diff line change
Expand Up @@ -487,3 +487,14 @@ @article{anderson2014fault
year={2014},
publisher={APS}
}

@article{lin2024quantum,
title={Quantum two-block group algebra codes},
author={Lin, Hsiang-Ku and Pryadko, Leonid P},
journal={Physical Review A},
volume={109},
number={2},
pages={022407},
year={2024},
publisher={APS}
}
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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254 changes: 254 additions & 0 deletions test/test_ecc_2bga_group_presentation.jl
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@@ -0,0 +1,254 @@
@testitem "ECC 2BGA abelian and non-abelian groups via group presentation" 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 1 Block 1" begin
# [[72, 8, 9]]
l = 36
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^l])
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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l
@test describe(G) == "C$l"
@test code_n(c) == 72 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(l, 2)) is C₃₆, cross-check it with G
@test small_group_identification(G) == (order(G), 2)
end

@testset "Reproduce Table 1 Block 2" begin
# [[72, 8, 9]]
l = 9
m = 4
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^m, r^l, 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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l*m
@test describe(G) == "C$l : C$m"
@test code_n(c) == 72 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(l*m, 1)) is C₉ x C₄, cross-check it with G
@test small_group_identification(G) == (order(G), 1)

# [[80, 8, 10]]
l = 5
m = 8
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^l, r^m, 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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l*m
@test describe(G) == "C$l : C$m"
@test code_n(c) == 80 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(l*m, 1)) is C₅ x C₈, cross-check it with G
@test small_group_identification(G) == (order(G), 1)
end

@testset "Reproduce Table 1 Block 3" begin
# [[54, 6, 9]]
l = 27
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^l])
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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l
@test describe(G) == "C$l"
@test code_n(c) == 54 && code_k(c) == 6
# Oscar.describe(Oscar.small_group(l, 1)) is C₂₇, cross-check it with G
@test small_group_identification(G) == (order(G), 1)

# [[60, 6, 10]]
l = 30
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^l])
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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l
@test describe(G) == "C$l"
@test code_n(c) == 60 && code_k(c) == 6
# Oscar.describe(Oscar.small_group(l, 4)) is C₃₀, cross-check it with G
@test small_group_identification(G) == (order(G), 4)

# [[70, 8, 10]]
l = 35
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^l])
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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l
@test describe(G) == "C$l"
@test code_n(c) == 70 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(l, 1)) is C₃₅, cross-check it with G
@test small_group_identification(G) == (order(G), 1)

# [[72, 8, 10]]
l = 36
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^l])
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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l
@test describe(G) == "C$l"
@test code_n(c) == 72 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(l, 2)) is C₃₆, cross-check it with G
@test small_group_identification(G) == (order(G), 2)

# [[72, 10, 9]]
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^l])
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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l
@test describe(G) == "C$l"
@test code_n(c) == 72 && code_k(c) == 10
# Oscar.describe(Oscar.small_group(l, 2)) is C₃₆, cross-check it with G
@test small_group_identification(G) == (order(G), 2)
end

@testset "Reproduce Table 1 Block 4" begin
# [[72, 8, 9]]
l = 9
m = 4
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^m, r^l, 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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l*m
@test describe(G) == "C$l : C$m"
@test code_n(c) == 72 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(l*m, 1)) is C₉ x C₄, cross-check it with G
@test small_group_identification(G) == (order(G), 1)

# [[80, 8, 10]]
l = 5
m = 8
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^l, r^m, 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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l*m
@test describe(G) == "C$l : C$m"
@test code_n(c) == 80 && code_k(c) == 8
# Oscar.describe(Oscar.small_group(l*m, 1)) is C₅ x C₈, cross-check it with G
@test small_group_identification(G) == (order(G), 1)

# [[96, 8, 12]]
l = 3
m = 16
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^l, r^m, 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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l*m
@test describe(G) == "C$l : C$m"
@test code_n(c) == 96 && code_k(c) == 6
# Oscar.describe(Oscar.small_group(l*m, 1)) is C₃ x C₁₆, cross-check it with G
@test small_group_identification(G) == (order(G), 1)

# [[84, 10, 9]]
l = 7
m = 3
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^m, 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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == 2*l*m
@test describe(G) == "C$l x S$m"
@test code_n(c) == 84 && code_k(c) == 10
# Oscar.describe(Oscar.small_group(2*l*m, 3)) is C₇ x S₃, cross-check it with G
@test small_group_identification(G) == (order(G), 3)

# [[96, 6, 12]]
l = 12
m = 4
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^m, r^l, 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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l*m
@test describe(G) == "C$l : C$m"
@test code_n(c) == 96 && code_k(c) == 6
# Oscar.describe(Oscar.small_group(l*m, 13)) is C₁₂ x C₄, cross-check it with G
@test small_group_identification(G) == (order(G), 13)

# [[96, 12, 10]]
l = 2
m = 3
n = 8
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^(l*m), r^n, 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]
c = get_code(a_elts, b_elts, F2G)
@test order(G) == l*m*n
@test describe(G) == "C$l x (C$m : C$n)"
@test code_n(c) == 96 && code_k(c) == 12
# Oscar.describe(Oscar.small_group(l*m*n, 9)) is C₂ x (C₃ : C₈), cross-check it with G
@test small_group_identification(G) == (order(G), 9)
end
end
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