using Hecke's Small Groups without any extra relations for single cyc…

…lic groups

ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl

@@ -5,7 +5,7 @@ using DocStringExtensions
 import QuantumClifford, LinearAlgebra
 import Hecke: Group, GroupElem, AdditiveGroup, AdditiveGroupElem,
     GroupAlgebra, GroupAlgebraElem, FqFieldElem, representation_matrix, dim, base_ring,
-    multiplication_table, coefficients, abelian_group, group_algebra
+    multiplication_table, coefficients, abelian_group, group_algebra, small_group, one
 import Nemo
 import Nemo: characteristic, matrix_repr, GF, ZZ, lift
 

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -153,8 +153,43 @@ code_s(c::LPCode) = size(c.repr(zero(c.GA)), 1) * (size(c.A, 1) * size(c.B, 1) +
 Two-block group algebra (2GBA) codes, which are a special case of lifted product codes
 from two group algebra elements `a` and `b`, used as `1x1` base matrices.
 
+# Example
+
+An abelian `[[60, 6, 10]]` 2BGA code  of order `l = 30` with group ID `4`, represented
+by the group presentation `⟨r | r³⁰⟩`, constructed via `Hecke.small_group`.
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens, small_group, one; # hide
+
+julia> l = 30;
+
+julia> group_id = 4;
+
+julia> G = small_group(l, group_id);
+
+julia> GA = group_algebra(GF(2), G);
+
+julia> r = prod(gens(GA));
+
+julia> r^30  ==  1 # presentation ⟨r|r³⁰⟩ satisfied 
+true
+
+julia> a_elts = [one(G), r^10, r^6, r^13];
+
+julia> b_elts = [one(G), r^25, r^16, r^12];
+
+julia> a = sum(GA(x) for x in a_elts);
+
+julia> b = sum(GA(x) for x in b_elts);
+
+julia> c = two_block_group_algebra_codes(a,b);
+
+julia> code_n(c), code_k(c) 
+(60, 6)
+```
+
 See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref)
-""" # TODO doctest example
+"""
 function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
     A = reshape([a], (1, 1))
     B = reshape([b], (1, 1))

test/test_ecc_small_groups.jl

@@ -0,0 +1,91 @@
+@testitem "ECC 2BGA Hecke Small Groups" begin
+    using Hecke: group_algebra, GF, abelian_group, gens, quo, one, GroupAlgebra, small_group
+    using QuantumClifford.ECC
+    using QuantumClifford.ECC: code_k, code_n, two_block_group_algebra_codes
+
+    @testset "Hecke Small Groups without extra relations for single cyclic groups" begin
+        # [[72, 8, 9]]
+        l = 36
+        group_id = 2
+        G = small_group(l, group_id)
+        GA = group_algebra(GF(2), G)
+        r = prod(gens(GA))
+        @test r^36  ==  1 # presentation ⟨r|r³⁶⟩ satisfied
+        a_elts = [one(G), r^28]
+        b_elts = [one(G), r, r^18, r^12, r^29, r^14]
+        a = sum(GA(x) for x in a_elts)
+        b = sum(GA(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a,b)
+        @test code_n(c) == 72 && code_k(c) == 8
+
+        # [[54, 6, 9]]
+        l = 27
+        group_id = 1
+        G = small_group(l, group_id)
+        GA = group_algebra(GF(2), G)
+        r = prod(gens(GA))
+        @test r^27  ==  1 # presentation ⟨r|r²⁷⟩ satisfied
+        a_elts = [one(G), r, r^3, r^7]
+        b_elts = [one(G), r, r^12, r^19]
+        a = sum(GA(x) for x in a_elts)
+        b = sum(GA(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a,b)
+        @test code_n(c) == 54 && code_k(c) == 6
+
+        # [[60, 6, 10]]
+        l = 30
+        group_id = 4
+        G = small_group(l, group_id)
+        GA = group_algebra(GF(2), G)
+        r = prod(gens(GA))
+        @test r^30  ==  1 # presentation ⟨r|r³⁰⟩ satisfied
+        a_elts = [one(G), r^10, r^6, r^13]
+        b_elts = [one(G), r^25, r^16, r^12]
+        a = sum(GA(x) for x in a_elts)
+        b = sum(GA(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a,b)
+        @test code_n(c) == 60 && code_k(c) == 6
+
+        # [[70, 8, 10]]
+        l = 35
+        group_id = 1
+        G = small_group(l, group_id)
+        GA = group_algebra(GF(2), G)
+        r = prod(gens(GA))
+        @test r^35  ==  1 # presentation ⟨r|r³⁵⟩ satisfied
+        a_elts = [one(G), r^15, r^16, r^18]
+        b_elts = [one(G), r, r^24, r^27]
+        a = sum(GA(x) for x in a_elts)
+        b = sum(GA(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a,b)
+        @test code_n(c) == 70 && code_k(c) == 8
+
+        # [[72, 8, 10]]
+        l = 36
+        group_id = 2
+        G = small_group(l, group_id)
+        GA = group_algebra(GF(2), G)
+        r = prod(gens(GA))
+        @test r^36  ==  1 # presentation ⟨r|r³⁶⟩ satisfied
+        a_elts = [one(G), r^9, r^28, r^31]
+        b_elts = [one(G), r, r^21, r^34]
+        a = sum(GA(x) for x in a_elts)
+        b = sum(GA(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a,b)
+        @test code_n(c) == 72 && code_k(c) == 8
+
+        # [[72, 10, 9]]
+        l = 36
+        group_id = 2
+        G = small_group(l, group_id)
+        GA = group_algebra(GF(2), G)
+        r = prod(gens(GA))
+        @test r^36  ==  1 # presentation ⟨r|r³⁶⟩ satisfied
+        a_elts = [one(G), r^9, r^28, r^13]
+        b_elts = [one(G), r, r^3, r^22]
+        a = sum(GA(x) for x in a_elts)
+        b = sum(GA(x) for x in b_elts)
+        c = two_block_group_algebra_codes(a,b)
+        @test code_n(c) == 72 && code_k(c) == 10
+    end
+end