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Group Presentation via Oscar Free Group
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@Fe-r-oz
Fe-r-oz committed Oct 18, 2024
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42 changes: 0 additions & 42 deletions ext/QuantumCliffordHeckeExt/lifted_product.jl
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Expand Up @@ -161,48 +161,6 @@ function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
LPCode(A, B)
end

"""
[[72, 8, 9]] 2BGA code from Table 1 of [lin2024quantum](@cite) with direct product
of `C₉ x C₄`.
```jldoctest
julia> c = two_block_group_algebra_codes([0, 28], [0, 9, 18, 12, 29, 14], (36, 2));
julia> code_n(c), code_k(c)
(72, 8)
```
"""
function two_block_group_algebra_codes(a_shifts::Array{Int}, b_shifts::Array{Int}, sg::Tuple{Int,Int})
m, i = sg
g1 = small_group(m, i)
GA = group_algebra(GF(2), g1)
a = sum(GA[n%dim(GA)+1] for n in a_shifts)
b = sum(GA[n%dim(GA)+1] for n in b_shifts)
two_block_group_algebra_codes(a, b)
end

"""
[[48, 8, 6]] 2BGA code from Table 3 of [lin2024quantum](@cite) with dihedral group of
order `l = 12`.
```jldoctest
julia> c = two_block_group_algebra_codes([0, 10], [0, 8, 9, 4, 2, 5], (12, 4), (2, 1));
julia> code_n(c), code_k(c)
(48, 8)
```
"""
function two_block_group_algebra_codes(a_shifts::Array{Int}, b_shifts::Array{Int}, sg1::Tuple{Int,Int}, sg2::Tuple{Int,Int})
m, i = sg1
l, i = sg2
g1 = small_group(m, i)
g2 = small_group(l, i)
GA = group_algebra(GF(2), direct_product(g1, g2))
a = sum(GA[n%dim(GA)+1] for n in a_shifts)
b = sum(GA[n%dim(GA)+1] for n in b_shifts)
two_block_group_algebra_codes(a, b)
end

"""
Generalized bicycle codes, which are a special case of 2GBA codes (and therefore of lifted product codes).
Here the group is chosen as the cyclic group of order `l`,
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253 changes: 253 additions & 0 deletions test/test_ecc_nonabelin2bga.jl
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@testitem "ECC 2BGA lin2024quantum" begin
import Hecke: group_algebra, GF, abelian_group, gens, quo, one
using QuantumClifford.ECC: LPCode, code_k, code_n, two_block_group_algebra_codes
using Oscar: free_group, small_group_identification, describe

@testset "Reproduce Table 1 Block 1" begin
# [[72, 8, 9]]
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^36])
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]
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) == "C36"
@test small_group_identification(G) == (36, 2)
@test code_n(c) == 72 && code_k(c) == 8
end

@testset "Reproduce Table 1 Block 2" begin
# [[72, 8, 9]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^4, r^9, 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]
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) == "C9 : C4"
@test small_group_identification(G) == (36, 1)
@test code_n(c) == 72 && code_k(c) == 8

# [[80, 8, 10]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^5, r^8, 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]
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) == "C5 : C8"
@test small_group_identification(G) == (40, 1)
@test code_n(c) == 80 && code_k(c) == 8

# [[96, 8, 12]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^6, r^8, (r*s)^8])
F2G = group_algebra(GF(2), G)
r, s = gens(G)
a_elts = [one(G), s*r^2]
b_elts = [one(G), r, s^3, s^4, s^2 * r^5, s^4 * 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) == "C5 : C8"
@test small_group_identification(G) == (40, 1)
@test code_n(c) == 80 && code_k(c) == 8
end

@testset "Reproduce Table 1 Block 3" begin
# [[54, 6, 9]]
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^27])
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]
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) == "C27"
@test small_group_identification(G) == (27, 1)
@test code_n(c) == 54 && code_k(c) == 6

# [[60, 6, 10]]
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^30])
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]
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) == "C30"
@test small_group_identification(G) == (30, 4)
@test code_n(c) == 60 && code_k(c) == 6

# [[70, 8, 10]]
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^35])
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]
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) == "C35"
@test small_group_identification(G) == (35, 1)
@test code_n(c) == 70 && code_k(c) == 8

# [[72, 8, 10]]
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^36])
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]
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) == "C36"
@test small_group_identification(G) == (36, 2)
@test code_n(c) == 72 && code_k(c) == 8

# [[72, 10, 9]]
F = free_group(["r"])
r = gens(F)[1]
G, = quo(F, [r^36])
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]
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) == "C36"
@test small_group_identification(G) == (36, 2)
@test code_n(c) == 72 && code_k(c) == 10
end

@testset "Reproduce Table 1 Block 4" begin
# [[72, 8, 9]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^4, r^9, 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]
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) == "C9 : C4"
@test small_group_identification(G) == (36, 1)
@test code_n(c) == 72 && code_k(c) == 8

# [[80, 8, 10]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^5, r^8, 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]
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) == "C5 : C8"
@test small_group_identification(G) == (40, 1)
@test code_n(c) == 80 && code_k(c) == 8

# [[96, 8, 12]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^3, r^16, 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]
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) == "C3 : C16"
@test small_group_identification(G) == (48, 1)
@test code_n(c) == 96 && code_k(c) == 6

# [[80, 9, 9]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^4, r^10, (r*s)^2])
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]
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) == "C7 x S3"
@test small_group_identification(G) == (42, 3)
@test code_n(c) == 84 && code_k(c) == 10

# [[84, 10, 9]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^3, 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]
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) == "C7 x S3"
@test small_group_identification(G) == (42, 3)
@test code_n(c) == 84 && code_k(c) == 10

# [[96, 6, 12]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^4, r^12, 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]
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) == "C12 : C4"
@test small_group_identification(G) == (48, 13)
@test code_n(c) == 96 && code_k(c) == 6

# [[96, 12, 10]]
F = free_group(["r", "s"])
r, s = gens(F)
G, = quo(F, [s^6, r^8, 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]
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) == "C2 x (C3 : C8)"
@test small_group_identification(G) == (48, 9)
@test code_n(c) == 96 && code_k(c) == 12
end
end

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