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‎test/test_ecc_2bga_group_presentation.jl‎

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@testitem "ECC 2BGA abelian and non-abelian groups via group presentation" begin
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using Nemo: FqFieldElem
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using Hecke: group_algebra, GF, abelian_group, gens, quo, one, GroupAlgebra
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using QuantumClifford.ECC
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using QuantumClifford.ECC: code_k, code_n, two_block_group_algebra_codes
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using Oscar: free_group, small_group_identification, describe, order, FPGroupElem, FPGroup, FPGroupElem
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function get_code(a_elts::Vector{FPGroupElem}, b_elts::Vector{FPGroupElem}, F2G::GroupAlgebra{FqFieldElem, FPGroup, FPGroupElem})
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a = sum(F2G(x) for x in a_elts)
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b = sum(F2G(x) for x in b_elts)
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c = two_block_group_algebra_codes(a,b)
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return c
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end
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@testset "Reproduce Table 1 Block 1" begin
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# [[72, 8, 9]]
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l = 36
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^l])
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F2G = group_algebra(GF(2), G)
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r = gens(G)[1]
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a_elts = [one(G), r^28]
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b_elts = [one(G), r, r^18, r^12, r^29, r^14]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l
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@test describe(G) == "C$l"
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@test code_n(c) == 72 && code_k(c) == 8
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# Oscar.describe(Oscar.small_group(l, 2)) is C₃₆, cross-check it with G
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@test small_group_identification(G) == (order(G), 2)
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end
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@testset "Reproduce Table 1 Block 2" begin
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# [[72, 8, 9]]
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l = 9
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m = 4
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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^m, r^l, s^(-1)*r*s*r])
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F2G = group_algebra(GF(2), G)
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r, s = gens(G)
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a_elts = [one(G), r]
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b_elts = [one(G), s, r^6, s^3 * r, s * r^7, s^3 * r^5]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l*m
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@test describe(G) == "C$l : C$m"
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@test code_n(c) == 72 && code_k(c) == 8
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# Oscar.describe(Oscar.small_group(l*m, 1)) is C₉ x C₄, cross-check it with G
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@test small_group_identification(G) == (order(G), 1)
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# [[80, 8, 10]]
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l = 5
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m = 8
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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^l, r^m, r^(-1)*s*r*s])
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F2G = group_algebra(GF(2), G)
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r, s = gens(G)
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a_elts = [one(G), s*r^4]
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b_elts = [one(G), r, r^2, s, s^3 * r, s^2 * r^6]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l*m
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@test describe(G) == "C$l : C$m"
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@test code_n(c) == 80 && code_k(c) == 8
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# Oscar.describe(Oscar.small_group(l*m, 1)) is C₅ x C₈, cross-check it with G
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@test small_group_identification(G) == (order(G), 1)
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end
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@testset "Reproduce Table 1 Block 3" begin
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# [[54, 6, 9]]
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l = 27
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^l])
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F2G = group_algebra(GF(2), G)
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r = gens(G)[1]
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a_elts = [one(G), r, r^3, r^7]
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b_elts = [one(G), r, r^12, r^19]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l
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@test describe(G) == "C$l"
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@test code_n(c) == 54 && code_k(c) == 6
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# Oscar.describe(Oscar.small_group(l, 1)) is C₂₇, cross-check it with G
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@test small_group_identification(G) == (order(G), 1)
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# [[60, 6, 10]]
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l = 30
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^l])
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F2G = group_algebra(GF(2), G)
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r = gens(G)[1]
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a_elts = [one(G), r^10, r^6, r^13]
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b_elts = [one(G), r^25, r^16, r^12]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l
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@test describe(G) == "C$l"
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@test code_n(c) == 60 && code_k(c) == 6
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# Oscar.describe(Oscar.small_group(l, 4)) is C₃₀, cross-check it with G
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@test small_group_identification(G) == (order(G), 4)
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# [[70, 8, 10]]
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l = 35
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^l])
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F2G = group_algebra(GF(2), G)
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r = gens(G)[1]
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a_elts = [one(G), r^15, r^16, r^18]
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b_elts = [one(G), r, r^24, r^27]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l
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@test describe(G) == "C$l"
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@test code_n(c) == 70 && code_k(c) == 8
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# Oscar.describe(Oscar.small_group(l, 1)) is C₃₅, cross-check it with G
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@test small_group_identification(G) == (order(G), 1)
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# [[72, 8, 10]]
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l = 36
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^l])
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F2G = group_algebra(GF(2), G)
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r = gens(G)[1]
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a_elts = [one(G), r^9, r^28, r^31]
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b_elts = [one(G), r, r^21, r^34]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l
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@test describe(G) == "C$l"
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@test code_n(c) == 72 && code_k(c) == 8
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# Oscar.describe(Oscar.small_group(l, 2)) is C₃₆, cross-check it with G
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@test small_group_identification(G) == (order(G), 2)
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# [[72, 10, 9]]
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^l])
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F2G = group_algebra(GF(2), G)
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r = gens(G)[1]
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a_elts = [one(G), r^9, r^28, r^13]
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b_elts = [one(G), r, r^3, r^22]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l
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@test describe(G) == "C$l"
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@test code_n(c) == 72 && code_k(c) == 10
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# Oscar.describe(Oscar.small_group(l, 2)) is C₃₆, cross-check it with G
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@test small_group_identification(G) == (order(G), 2)
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end
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@testset "Reproduce Table 1 Block 4" begin
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# [[72, 8, 9]]
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l = 9
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m = 4
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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^m, r^l, s^(-1)*r*s*r])
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F2G = group_algebra(GF(2), G)
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r, s = gens(G)
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a_elts = [one(G), s, r, s*r^6]
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b_elts = [one(G), s^2*r, s^2*r^6, r^2]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l*m
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@test describe(G) == "C$l : C$m"
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@test code_n(c) == 72 && code_k(c) == 8
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# Oscar.describe(Oscar.small_group(l*m, 1)) is C₉ x C₄, cross-check it with G
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@test small_group_identification(G) == (order(G), 1)
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# [[80, 8, 10]]
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l = 5
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m = 8
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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^l, r^m, r^(-1)*s*r*s])
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F2G = group_algebra(GF(2), G)
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r, s = gens(G)
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a_elts = [one(G), r, s, s^3*r^5]
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b_elts = [one(G), r^2, s*r^4, s^3*r^2]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l*m
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@test describe(G) == "C$l : C$m"
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@test code_n(c) == 80 && code_k(c) == 8
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# Oscar.describe(Oscar.small_group(l*m, 1)) is C₅ x C₈, cross-check it with G
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@test small_group_identification(G) == (order(G), 1)
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# [[96, 8, 12]]
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l = 3
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m = 16
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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^l, r^m, r^(-1)*s*r*s])
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F2G = group_algebra(GF(2), G)
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r, s = gens(G)
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a_elts = [one(G), r, s, r^14]
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b_elts = [one(G), r^2, s*r^4, r^11]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l*m
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@test describe(G) == "C$l : C$m"
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@test code_n(c) == 96 && code_k(c) == 6
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# Oscar.describe(Oscar.small_group(l*m, 1)) is C₃ x C₁₆, cross-check it with G
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@test small_group_identification(G) == (order(G), 1)
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# [[84, 10, 9]]
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l = 7
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m = 3
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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^m, r^14, r^(-1)*s*r*s])
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F2G = group_algebra(GF(2), G)
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r, s = gens(G)
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a_elts = [one(G), r^7, r^8, s*r^10]
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b_elts = [one(G), s, r^5, s^2*r^13]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == 2*l*m
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@test describe(G) == "C$l x S$m"
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@test code_n(c) == 84 && code_k(c) == 10
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# Oscar.describe(Oscar.small_group(2*l*m, 3)) is C₇ x S₃, cross-check it with G
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@test small_group_identification(G) == (order(G), 3)
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# [[96, 6, 12]]
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l = 12
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m = 4
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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^m, r^l, s^(-1)*r*s*r])
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F2G = group_algebra(GF(2), G)
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r, s = gens(G)
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a_elts = [one(G), s, r^9, s * r]
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b_elts = [one(G), s^2 * s^9, r^7, r^2]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l*m
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@test describe(G) == "C$l : C$m"
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@test code_n(c) == 96 && code_k(c) == 6
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# Oscar.describe(Oscar.small_group(l*m, 13)) is C₁₂ x C₄, cross-check it with G
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@test small_group_identification(G) == (order(G), 13)
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# [[96, 12, 10]]
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l = 2
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m = 3
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n = 8
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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^(l*m), r^n, r^(-1)*s*r*s])
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F2G = group_algebra(GF(2), G)
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r, s = gens(G)
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a_elts = [one(G), r, s^3 * r^2, s^2 * r^3]
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b_elts = [one(G), r, s^4 * r^6, s^5 * r^3]
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c = get_code(a_elts, b_elts, F2G)
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@test order(G) == l*m*n
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@test describe(G) == "C$l x (C$m : C$n)"
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@test code_n(c) == 96 && code_k(c) == 12
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# Oscar.describe(Oscar.small_group(l*m*n, 9)) is C₂ x (C₃ : C₈), cross-check it with G
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@test small_group_identification(G) == (order(G), 9)
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end
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end

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