add consistency check to verify F-linear regular representation for G…

…roup Algebra elements

ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl

@@ -13,6 +13,7 @@ import QuantumClifford.ECC: AbstractECC, CSS, ClassicalCode,
     hgp, code_k, code_n, code_s, iscss, parity_checks, parity_checks_x, parity_checks_z, parity_checks_xz,
     two_block_group_algebra_codes, generalized_bicycle_codes, bicycle_codes
 
+include("util.jl")
 include("types.jl")
 include("lifted.jl")
 include("lifted_product.jl")

ext/QuantumCliffordHeckeExt/util.jl

@@ -0,0 +1,17 @@
+""""
+Verifies that the specified group elements constitute a regular F-linear representation
+of the group G by checking the properties L(a)L(b) = L(ab) and R(a)R(b) = R(ba) for any
+elements a, b in the group algebra F[G].
+"""
+function check_repr_regular_linear(GA::GroupAlgebra)
+    a, b = rand(GA), rand(GA)
+    # L(a)L(b) = L(ab)
+    L_a = representation_matrix(a)
+    L_b = representation_matrix(b)
+    L_ab = representation_matrix(a*b)
+    # R(a)R(b) = R(ba)
+    R_a = representation_matrix(a, :right)
+    R_b = representation_matrix(b, :right)
+    R_ba = representation_matrix(b*a, :right)
+    return L_a * L_b == L_ab && R_a * R_b == R_ba
+end

src/ecc/codes/util.jl

@@ -6,3 +6,6 @@ function hgp(h₁,h₂)
     hz = hcat(kron(LinearAlgebra.I(n₁), h₂), kron(h₁', LinearAlgebra.I(r₂)))
     hx, hz
 end
+
+"""Implemented in a package extension with Hecke."""
+function check_repr_regular_linear end

test/test_ecc_base.jl

@@ -1,6 +1,7 @@
 using Test
 using QuantumClifford
 using QuantumClifford.ECC
+using QuantumClifford.ECC: check_repr_regular_linear
 using InteractiveUtils
 
 import Nemo: GF
@@ -46,6 +47,7 @@ other_lifted_product_codes = []
 # [[882, 24, d≤24]] code from (B1) in Appendix B of [panteleev2021degenerate](@cite)
 l = 63
 GA = group_algebra(GF(2), abelian_group(l))
+@test check_repr_regular_linear(GA)
 A = zeros(GA, 7, 7)
 x = gens(GA)[]
 A[LinearAlgebra.diagind(A)] .= x^27