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‎ext/QuantumCliffordHeckeExt/lifted_product.jl‎

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@@ -161,48 +161,6 @@ function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
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LPCode(A, B)
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end
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"""
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[[72, 8, 9]] 2BGA code from Table 1 of [lin2024quantum](@cite) with direct product
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of `C₉ x C₄`.
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```jldoctest
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julia> c = two_block_group_algebra_codes([0, 28], [0, 9, 18, 12, 29, 14], (36, 2));
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julia> code_n(c), code_k(c)
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(72, 8)
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```
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"""
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function two_block_group_algebra_codes(a_shifts::Array{Int}, b_shifts::Array{Int}, sg::Tuple{Int,Int})
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m, i = sg
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g1 = small_group(m, i)
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GA = group_algebra(GF(2), g1)
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a = sum(GA[n%dim(GA)+1] for n in a_shifts)
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b = sum(GA[n%dim(GA)+1] for n in b_shifts)
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two_block_group_algebra_codes(a, b)
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end
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"""
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[[48, 8, 6]] 2BGA code from Table 3 of [lin2024quantum](@cite) with dihedral group of
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order `l = 12`.
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```jldoctest
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julia> c = two_block_group_algebra_codes([0, 10], [0, 8, 9, 4, 2, 5], (12, 4), (2, 1));
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julia> code_n(c), code_k(c)
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(48, 8)
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```
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"""
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function two_block_group_algebra_codes(a_shifts::Array{Int}, b_shifts::Array{Int}, sg1::Tuple{Int,Int}, sg2::Tuple{Int,Int})
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m, i = sg1
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l, i = sg2
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g1 = small_group(m, i)
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g2 = small_group(l, i)
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GA = group_algebra(GF(2), direct_product(g1, g2))
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a = sum(GA[n%dim(GA)+1] for n in a_shifts)
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b = sum(GA[n%dim(GA)+1] for n in b_shifts)
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two_block_group_algebra_codes(a, b)
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end
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"""
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Generalized bicycle codes, which are a special case of 2GBA codes (and therefore of lifted product codes).
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Here the group is chosen as the cyclic group of order `l`,

‎test/test_ecc_nonabelin2bga.jl‎

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@testitem "ECC 2BGA lin2024quantum" begin
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import Hecke: group_algebra, GF, abelian_group, gens, quo, one
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using QuantumClifford.ECC: LPCode, code_k, code_n, two_block_group_algebra_codes
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using Oscar: free_group, small_group_identification, describe
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@testset "Reproduce Table 1 Block 1" begin
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# [[72, 8, 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^36])
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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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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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@test describe(G) == "C36"
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@test small_group_identification(G) == (36, 2)
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@test code_n(c) == 72 && code_k(c) == 8
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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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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^4, r^9, 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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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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@test describe(G) == "C9 : C4"
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@test small_group_identification(G) == (36, 1)
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@test code_n(c) == 72 && code_k(c) == 8
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# [[80, 8, 10]]
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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^5, r^8, 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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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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@test describe(G) == "C5 : C8"
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@test small_group_identification(G) == (40, 1)
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@test code_n(c) == 80 && code_k(c) == 8
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# [[96, 8, 12]]
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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^6, r^8, (r*s)^8])
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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^2]
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b_elts = [one(G), r, s^3, s^4, s^2 * r^5, s^4 * r^6]
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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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@test describe(G) == "C5 : C8"
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@test small_group_identification(G) == (40, 1)
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@test code_n(c) == 80 && code_k(c) == 8
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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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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^27])
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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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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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@test describe(G) == "C27"
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@test small_group_identification(G) == (27, 1)
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@test code_n(c) == 54 && code_k(c) == 6
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# [[60, 6, 10]]
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^30])
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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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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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@test describe(G) == "C30"
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@test small_group_identification(G) == (30, 4)
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@test code_n(c) == 60 && code_k(c) == 6
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# [[70, 8, 10]]
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^35])
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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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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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@test describe(G) == "C35"
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@test small_group_identification(G) == (35, 1)
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@test code_n(c) == 70 && code_k(c) == 8
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# [[72, 8, 10]]
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F = free_group(["r"])
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r = gens(F)[1]
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G, = quo(F, [r^36])
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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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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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@test describe(G) == "C36"
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@test small_group_identification(G) == (36, 2)
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@test code_n(c) == 72 && code_k(c) == 8
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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^36])
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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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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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@test describe(G) == "C36"
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@test small_group_identification(G) == (36, 2)
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@test code_n(c) == 72 && code_k(c) == 10
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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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F = free_group(["r", "s"])
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r, s = gens(F)
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G, = quo(F, [s^4, r^9, 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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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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@test describe(G) == "C9 : C4"
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@test small_group_identification(G) == (36, 1)
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@test code_n(c) == 72 && code_k(c) == 8
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# [[80, 8, 10]]
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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^5, r^8, 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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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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@test describe(G) == "C5 : C8"
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@test small_group_identification(G) == (40, 1)
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@test code_n(c) == 80 && code_k(c) == 8
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# [[96, 8, 12]]
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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^3, r^16, 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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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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@test describe(G) == "C3 : C16"
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@test small_group_identification(G) == (48, 1)
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@test code_n(c) == 96 && code_k(c) == 6
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# [[80, 9, 9]]
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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^4, r^10, (r*s)^2])
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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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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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@test describe(G) == "C7 x S3"
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@test small_group_identification(G) == (42, 3)
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@test code_n(c) == 84 && code_k(c) == 10
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# [[84, 10, 9]]
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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^3, 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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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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@test describe(G) == "C7 x S3"
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@test small_group_identification(G) == (42, 3)
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@test code_n(c) == 84 && code_k(c) == 10
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# [[96, 6, 12]]
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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^4, r^12, 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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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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@test describe(G) == "C12 : C4"
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@test small_group_identification(G) == (48, 13)
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@test code_n(c) == 96 && code_k(c) == 6
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# [[96, 12, 10]]
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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^6, r^8, 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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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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@test describe(G) == "C2 x (C3 : C8)"
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@test small_group_identification(G) == (48, 9)
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@test code_n(c) == 96 && code_k(c) == 12
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end
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end

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