introducing non-abelian groups via Hecke's small group with multiplic…

…ation table

docs/src/references.bib

@@ -487,3 +487,14 @@ @article{anderson2014fault
   year={2014},
   publisher={APS}
 }
+
+@article{lin2024quantum,
+  title={Quantum two-block group algebra codes},
+  author={Lin, Hsiang-Ku and Pryadko, Leonid P},
+  journal={Physical Review A},
+  volume={109},
+  number={2},
+  pages={022407},
+  year={2024},
+  publisher={APS}
+}

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, direct_product
 import Nemo
 import Nemo: characteristic, matrix_repr, GF, ZZ, lift
 

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -150,6 +150,28 @@ function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
     LPCode(A, B)
 end
 
+"""
+[[48, 8, 6]] 2BGA code from Table 1 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`,