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Non-Abelian Dihedral Groups via Group Presentation
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@Fe-r-oz
Fe-r-oz committed Oct 18, 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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297 changes: 297 additions & 0 deletions test/test_ecc_dihedral2bga.jl
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@testitem "ECC 2BGA lin2024quantum" begin
using Hecke
using Hecke: group_algebra, GF, abelian_group, gens, quo, one
using QuantumClifford.ECC: LPCode, code_k, code_n
using Oscar: free_group, small_group_identification, describe, order

@testset "Reproduce Table 3 of lin2024quantum" begin
# [[24, 8, 3]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^6, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^4]
b_elts = [one(G), s*r^4, r^3, r^4, s*r^2, r]
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) == "D12"
@test order(G) == 12
@test small_group_identification(G) == (12, 4)
@test code_n(c) == 24 && code_k(c) == 8

# [[24, 12, 2]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^6, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^3]
b_elts = [one(G), s*r, r^3, r^4, s*r^4, r]
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) == "D12"
@test order(G) == 12
@test small_group_identification(G) == (12, 4)
@test code_n(c) == 24 && code_k(c) == 12

# [[32, 8, 4]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^8, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^2]
b_elts = [one(G), s*r^5, s*r^4, r^2, s*r^7, s*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) == "D16"
@test order(G) == 16
@test small_group_identification(G) == (16, 7)
@test code_n(c) == 32 && code_k(c) == 8

# [[32, 16, 2]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^8, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^4]
b_elts = [one(G), s*r^3, s*r^6, r^4, s*r^7, s*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) == "D16"
@test order(G) == 16
@test small_group_identification(G) == (16, 7)
@test code_n(c) == 32 && code_k(c) == 16

# [[36, 12, 3]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^9, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^3]
b_elts = [one(G), s, r, r^3, s*r^3, r^4]
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) == "D18"
@test order(G) == 18
@test small_group_identification(G) == (18, 1)
@test code_n(c) == 36 && code_k(c) == 12

# [[40, 8, 5]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^10, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^2]
b_elts = [one(G), s*r^4, r^5, r^2, s*r^6, r]
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) == "D20"
@test order(G) == 20
@test small_group_identification(G) == (20, 4)
@test code_n(c) == 40 && code_k(c) == 8

# [[40, 20, 2]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^10, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^5]
b_elts = [one(G), s*r^2, r^5, r^6, s*r^7, r]
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) == "D20"
@test order(G) == 20
@test small_group_identification(G) == (20, 4)
@test code_n(c) == 40 && code_k(c) == 20

# [[48, 8, 6]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^12, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^10]
b_elts = [one(G), s*r^8, r^9, r^4, s*r^2, 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) == "D24"
@test order(G) == 24
@test small_group_identification(G) == (24, 6)
@test code_n(c) == 48 && code_k(c) == 8

# [[48, 12, 4]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^12, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^3]
b_elts = [one(G), s*r^7, r^3, r^4, s*r^10, r^7]
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) == "D24"
@test order(G) == 24
@test small_group_identification(G) == (24, 6)
@test code_n(c) == 48 && code_k(c) == 12

# [[48, 16, 3]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^12, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^8]
b_elts = [one(G), s*r^8, r^9, r^8, s*r^4, 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) == "D24"
@test order(G) == 24
@test small_group_identification(G) == (24, 6)
@test code_n(c) == 48 && code_k(c) == 16

# [[48, 24, 2]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^12, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^6]
b_elts = [one(G), s*r^11, r^6, s*r^5, r, r^7]
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) == "D24"
@test order(G) == 24
@test small_group_identification(G) == (24, 6)
@test code_n(c) == 48 && code_k(c) == 24

# [[56, 8, 7]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^14, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^4]
b_elts = [one(G), s*r^11, r^7, s*r^5, r^12, r^9]
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) == "D28"
@test order(G) == 28
@test small_group_identification(G) == (28, 3)
@test code_n(c) == 56 && code_k(c) == 8

# [[56, 28, 2]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^14, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^7]
b_elts = [one(G), s*r^2, r^7, r^8, s*r^9, r]
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) == "D28"
@test order(G) == 28
@test small_group_identification(G) == (28, 3)
@test code_n(c) == 56 && code_k(c) == 28


# [[60, 12, 5]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^15, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^12]
b_elts = [one(G), s*r^14, r^5, r^12, s*r^11, 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) == "D30"
@test order(G) == 30
@test small_group_identification(G) == (30, 3)
@test code_n(c) == 60 && code_k(c) == 12

# [[60, 20, 3]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^15, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^5]
b_elts = [one(G), s*r^13, r^5, r^12, s*r^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) == "D30"
@test order(G) == 30
@test small_group_identification(G) == (30, 3)
@test code_n(c) == 60 && code_k(c) == 20

# [[64, 8, 8]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^16, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^6]
b_elts = [one(G), s*r^12, s*r^9, r^6, s, s*r]
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) == "D32"
@test order(G) == 32
@test small_group_identification(G) == (32, 18)
@test code_n(c) == 64 && code_k(c) == 8

# [[64, 16, 8]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^16, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^4]
b_elts = [one(G), s*r^10, s*r^3, r^4, s*r^14, s*r^7]
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) == "D32"
@test order(G) == 32
@test small_group_identification(G) == (32, 18)
@test code_n(c) == 64 && code_k(c) == 16

# [[64, 32, 2]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [r^16, s^2, (r*s)^2])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), r^8]
b_elts = [one(G), s*r^11, s*r^12, r^8, s*r^3, s*r^4]
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) == "D32"
@test order(G) == 32
@test small_group_identification(G) == (32, 18)
@test code_n(c) == 64 && code_k(c) == 32
end
end

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