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Group Presentation for Cyclic Group Cₘₕ = Cₘ × C₂
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‎test/Project.toml‎

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

‎test/test_ecc_2bga_cyclicgroups_presentation.jl‎

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@testitem "ECC 2BGA Table 2 via Presentation of Cyclic Groups" 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 2 lin2024quantum" begin
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# codes taken from Appendix C, Table 2 of [lin2024quantum](@cite)
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# [[16, 2, 4]]
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# m = 4
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F = free_group(["x", "s"])
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x, s = gens(F)
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G, = quo(F, [x^4, s^2, x * s * x^-1 * s^-1])
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F2G = group_algebra(GF(2), G)
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x, s = gens(G)
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a_elts = [one(G), x]
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b_elts = [one(G), x, s, x^2, s*x, s*x^3]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C4 x C2"
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@test order(G) == 8
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@test small_group_identification(G) == (8, 2)
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@test code_n(c) == 16 && code_k(c) == 2
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# [[16, 4, 4]]
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b_elts = [one(G), x, s, x^2, s*x, x^3]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C4 x C2"
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@test order(G) == 8
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@test small_group_identification(G) == (8, 2)
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@test code_n(c) == 16 && code_k(c) == 4
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# [[16, 8, 2]]
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a_elts = [one(G), s]
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b_elts = [one(G), x, s, x^2, s*x, x^2]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C4 x C2"
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@test order(G) == 8
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@test small_group_identification(G) == (8, 2)
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@test code_n(c) == 16 && code_k(c) == 8
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# [[24, 4, 5]]
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# m = 6
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F = free_group(["x", "s"])
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x, s = gens(F)
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G, = quo(F, [x^6, s^2, x * s * x^-1 * s^-1])
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F2G = group_algebra(GF(2), G)
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x, s = gens(G)
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a_elts = [one(G), x]
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b_elts = [one(G), x^3, s, x^4, x^2, s*x]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C6 x C2"
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@test order(G) == 12
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@test small_group_identification(G) == (12, 5)
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@test code_n(c) == 24 && code_k(c) == 4
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# [[24, 12, 2]]
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a_elts = [one(G), x^3]
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b_elts = [one(G), x^3, s, x^4, s * x^3, x]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C6 x C2"
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@test order(G) == 12
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@test small_group_identification(G) == (12, 5)
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@test code_n(c) == 24 && code_k(c) == 12
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# [[32, 8, 4]]
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# m = 8
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F = free_group(["x", "s"])
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x, s = gens(F)
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G, = quo(F, [x^8, s^2, x * s * x^-1 * s^-1])
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F2G = group_algebra(GF(2), G)
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x, s = gens(G)
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a_elts = [one(G), x^6]
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b_elts = [one(G), s * x^7, s * x^4, x^6, s * x^5, s * x^2]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C8 x C2"
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@test order(G) == 16
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@test small_group_identification(G) == (16, 5)
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@test code_n(c) == 32 && code_k(c) == 8
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# [[32, 16, 2]]
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a_elts = [one(G), s * x^4]
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b_elts = [one(G), s * x^7, s * x^4, x^6, x^3, s * x^2]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C8 x C2"
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@test order(G) == 16
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@test small_group_identification(G) == (16, 5)
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@test code_n(c) == 32 && code_k(c) == 16
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# [[40, 4, 8]]
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# m = 10
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F = free_group(["x", "s"])
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x, s = gens(F)
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G, = quo(F, [x^10, s^2, x * s * x^-1 * s^-1])
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F2G = group_algebra(GF(2), G)
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x, s = gens(G)
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a_elts = [one(G), x]
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b_elts = [one(G), x^5, x^6, s * x^6, x^7, s * x^3]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C10 x C2"
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@test order(G) == 20
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@test small_group_identification(G) == (20, 5)
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@test code_n(c) == 40 && code_k(c) == 4
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# [[40, 8, 5]]
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a_elts = [one(G), x^6]
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b_elts = [one(G), x^5, s, x^6 , x, s * x^2]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C10 x C2"
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@test order(G) == 20
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@test small_group_identification(G) == (20, 5)
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@test code_n(c) == 40 && code_k(c) == 8
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# [[40, 20, 2]]
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a_elts = [one(G), x^5]
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b_elts = [one(G), x^5, s, x^6, s * x^5, x]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C10 x C2"
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@test order(G) == 20
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@test small_group_identification(G) == (20, 5)
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@test code_n(c) == 40 && code_k(c) == 20
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# [[48, 8, 6]]
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# m = 12
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F = free_group(["x", "s"])
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x, s = gens(F)
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G, = quo(F, [x^12, s^2, x * s * x^-1 * s^-1])
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F2G = group_algebra(GF(2), G)
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x, s = gens(G)
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a_elts = [one(G), s * x^10]
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b_elts = [one(G), x^3, s * x^6, x^4, x^7, x^8]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C12 x C2"
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@test order(G) == 24
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@test small_group_identification(G) == (24, 9)
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@test code_n(c) == 48 && code_k(c) == 8
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# [[48, 12, 4]]
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a_elts = [one(G), x^3]
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b_elts = [one(G), x^3, s * x^6, x^4, s * x^9, x^7]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C12 x C2"
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@test order(G) == 24
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@test small_group_identification(G) == (24, 9)
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@test code_n(c) == 48 && code_k(c) == 12
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# [[48, 16, 3]]
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a_elts = [one(G), x^4]
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b_elts = [one(G), x^3, s * x^6, x^4, x^7, s * x^10]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C12 x C2"
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@test order(G) == 24
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@test small_group_identification(G) == (24, 9)
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@test code_n(c) == 48 && code_k(c) == 16
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# [[48, 24, 2]]
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a_elts = [one(G), s * x^6]
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b_elts = [one(G), x^3, s * x^6, x^4, s * x^9, s * x^10]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C12 x C2"
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@test order(G) == 24
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@test small_group_identification(G) == (24, 9)
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@test code_n(c) == 48 && code_k(c) == 24
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# [[56, 8, 7]]
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# m = 14
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F = free_group(["x", "s"])
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x, s = gens(F)
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G, = quo(F, [x^14, s^2, x * s * x^-1 * s^-1])
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F2G = group_algebra(GF(2), G)
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x, s = gens(G)
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a_elts = [one(G), x^8]
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b_elts = [one(G), x^7, s, x^8, x^9, s * x^4]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C14 x C2"
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@test order(G) == 28
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@test small_group_identification(G) == (28, 4)
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@test code_n(c) == 56 && code_k(c) == 8
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# [[56, 28, 2]]
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a_elts = [one(G), x^7]
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b_elts = [one(G), x^7, s, x^8, s * x^7, x]
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c = get_code(a_elts, b_elts, F2G)
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@test describe(G) == "C14 x C2"
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@test order(G) == 28
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@test small_group_identification(G) == (28, 4)
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@test code_n(c) == 56 && code_k(c) == 28
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end
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end

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