Hecke's Small Groups (small_group(l, Id)) without any extra relations for single cyclic 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, rand
+    multiplication_table, coefficients, abelian_group, group_algebra, rand, small_group
 import Nemo
 import Nemo: characteristic, matrix_repr, GF, ZZ, lift
 

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -265,6 +265,49 @@ julia> code_n(c), code_k(c)
 (108, 12)
 ```
 
+### Small Groups
+
+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(4,30)`. Note:
+Hecke's small groups are limited in scope and should only be used for single cyclic groups.
+
+```jldoctest sg
+julia> import Hecke: group_algebra, GF, abelian_group, gens, small_group; using QuantumClifford.ECC;
+
+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));
+```
+
+!!! note When using `Hecke.small_group`, it is essential to verify that the
+presentation for the single cyclic group is satisfied before proceeding with
+the code construction. This method serves as a workaround for creating small
+groups, specifically for single cyclic groups, using a group presentation with
+*no extra relations*, such as `⟨r | r³⁰⟩`. For the construction of *general*
+groups with specific group presentations, the only effective method is to use
+*finitely presented groups* (`Oscar.FPGroup`), which allow for defining direct
+products of two or more *general* groups—something not supported by Hecke.
+
+```jldoctest sg
+julia> r^30  ==  1
+true
+
+julia> A = 1 + r^10 + r^6  + r^13;
+
+julia> B = 1 + r^25 + r^16 + r^12;
+
+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), [`haah_cubic_codes`](@ref).
 """
 function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)

test/test_ecc_small_groups.jl

@@ -0,0 +1,79 @@
+@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 = 1 + r^28
+        B = 1 + r + r^18 + r^12 + r^29 + r^14
+        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 = 1 + r + r^3  + r^7
+        B = 1 + r + r^12 + r^19
+        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 = 1 + r^10 + r^6  + r^13
+        B = 1 + r^25 + r^16 + r^12
+        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 = 1 + r^15 + r^16 + r^18
+        B = 1 + r    + r^24 + r^27
+        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 = 1 + r^9 + r^28 + r^31
+        B = 1 + r   + r^21 + r^34
+        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 = 1 + r^9 + r^28 + r^13
+        B = 1 + r   + r^3  + r^22
+        c = two_block_group_algebra_codes(A,B)
+        @test code_n(c) == 72 && code_k(c) == 10
+    end
+end