Direct Products of Groups via Hecke for 2BGA codes to reproduce Table…

… 2 by H.K.Lin et al.

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/lifted_product.jl

@@ -142,8 +142,30 @@ 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.
 
+[[56, 28, 2]] 2BGA code from Table 2 of [lin2024quantum](@cite) with direct product
+of `C₄ x C₂`.
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens;
+
+julia> GA = group_algebra(GF(2), abelian_group([14,2]));
+
+julia> x = gens(GA)[1];
+
+julia> s = gens(GA)[2];
+
+julia> A = reshape([1 + x^7], (1, 1));
+
+julia> B = reshape([1 + x^7 + s + x^8 + s*x^7 + x], (1, 1));
+
+julia> c = LPCode(A,B);
+
+julia> code_n(c), code_k(c)
+(56, 28)
+```
+
 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_2bga.jl

@@ -0,0 +1,119 @@
+@testitem "ECC 2BGA" begin
+    using Hecke
+    using Hecke: group_algebra, GF, abelian_group, gens
+    using QuantumClifford.ECC: LPCode, code_k, code_n
+
+    @testset "Reproduce Table 2 lin2024quantum" begin
+        # codes taken from Table 2 of [lin2024quantum](@cite)
+
+        # m = 4
+        GA = group_algebra(GF(2), abelian_group([4,2]))
+        x = gens(GA)[1]
+        s = gens(GA)[2]
+        A = reshape([1 + x], (1, 1))
+        B = reshape([1 + x + s + x^2 + s*x + s*x^3], (1, 1))
+        c = LPCode(A,B)
+        # [[16, 2, 4]] 2BGA code
+        @test code_n(c) == 16 && code_k(c) == 2
+        A = reshape([1 + x], (1, 1))
+        B = reshape([1 + x + s + x^2 + s*x + x^3], (1, 1))
+        c = LPCode(A,B)
+        # [[16, 4, 4]] 2BGA code
+        @test code_n(c) == 16 && code_k(c) == 4
+        A = reshape([1 + s], (1, 1))
+        B = reshape([1 + x + s + x^2 + s*x + x^2], (1, 1))
+        c = LPCode(A,B)
+        # [[16, 8, 2]] 2BGA code
+        @test code_n(c) == 16 && code_k(c) == 8
+
+        # m = 6
+        GA = group_algebra(GF(2), abelian_group([6,2]))
+        x = gens(GA)[1]
+        s = gens(GA)[2]
+        A = reshape([1 + x], (1, 1))
+        B = reshape([1 + x^3 + s + x^4 + x^2 + s*x], (1, 1))
+        c = LPCode(A,B)
+        # [[24, 4, 5]] 2BGA code
+        @test code_n(c) == 24 && code_k(c) == 4
+        A = reshape([1 + x^3], (1, 1))
+        B = reshape([1 + x^3 + s + x^4 + s*x^3 + x], (1, 1))
+        c = LPCode(A,B)
+        # [[24, 12, 2]] 2BGA code
+        @test code_n(c) == 24 && code_k(c) == 12
+
+        # m = 8
+        GA = group_algebra(GF(2), abelian_group([8,2]))
+        x = gens(GA)[1]
+        s = gens(GA)[2]
+        A = reshape([1 + x^6], (1, 1))
+        B = reshape([1 + s*x^7 + s*x^4 + x^6 + s*x^5 + s*x^2], (1, 1))
+        c = LPCode(A,B)
+        # [[32, 8, 4]] 2BGA code
+        @test code_n(c) == 32 && code_k(c) == 8
+        A = reshape([1 + s*x^4], (1, 1))
+        B = reshape([1 + s*x^7 + s*x^4 + x^6 + x^3 + s*x^2], (1, 1))
+        c = LPCode(A,B)
+        # [[32, 16, 2]] 2BGA code
+        @test code_n(c) == 32 && code_k(c) == 16
+
+        # m = 10
+        GA = group_algebra(GF(2), abelian_group([10,2]))
+        x = gens(GA)[1]
+        s = gens(GA)[2]
+        A = reshape([1 + x], (1, 1))
+        B = reshape([1 + x^5 + x^6 + s*x^6 + x^7 + s*x^3], (1, 1))
+        c = LPCode(A,B)
+        # [[40, 4, 8]] 2BGA code
+        @test code_n(c) == 40 && code_k(c) == 4
+        A = reshape([1 + x^6], (1, 1))
+        B = reshape([1 + x^5 + s + x^6 + x + s*x^2], (1, 1))
+        c = LPCode(A,B)
+        # [[40, 8, 5]] 2BGA code
+        @test code_n(c) == 40 && code_k(c) == 8
+        A = reshape([1 + x^5], (1, 1))
+        B = reshape([1 + x^5 + s + x^6 + s*x^5 + x], (1, 1))
+        c = LPCode(A,B)
+        # [[40, 20, 2]] 2BGA code
+        @test code_n(c) == 40 && code_k(c) == 20
+
+        # m = 12
+        GA = group_algebra(GF(2), abelian_group([12,2]))
+        x = gens(GA)[1]
+        s = gens(GA)[2]
+        A = reshape([1 + s*x^10], (1, 1))
+        B = reshape([1 + x^3 + s*x^6 + x^4 + x^7 + x^8], (1, 1))
+        c = LPCode(A,B)
+        # [[48, 8, 6]] 2BGA code
+        @test code_n(c) == 48 && code_k(c) == 8
+        A = reshape([1 + x^3], (1, 1))
+        B = reshape([1 + x^3 + s*x^6 + x^4 + s*x^9 + x^7], (1, 1))
+        c = LPCode(A,B)
+        # [[48, 12, 4]] 2BGA code
+        @test code_n(c) == 48 && code_k(c) == 12
+        A = reshape([1 + x^4], (1, 1))
+        B = reshape([1 + x^3 + s*x^6 + x^4 + x^7 + s*x^10], (1, 1))
+        c = LPCode(A,B)
+        # [[48, 16, 3]] 2BGA code
+        @test code_n(c) == 48 && code_k(c) == 16
+        A = reshape([1 + s*x^6], (1, 1))
+        B = reshape([1 + x^3 + s*x^6 + x^4 + s*x^9 + s*x^10], (1, 1))
+        c = LPCode(A,B)
+        # [[48, 24, 2]] 2BGA code
+        @test code_n(c) == 48 && code_k(c) == 24
+
+        # m = 14
+        GA = group_algebra(GF(2), abelian_group([14,2]))
+        x = gens(GA)[1]
+        s = gens(GA)[2]
+        A = reshape([1 + x^8], (1, 1))
+        B = reshape([1 + x^7 + s + x^8 + x^9 + s*x^4], (1, 1))
+        c = LPCode(A,B)
+        # [[56, 8, 7]] 2BGA code
+        @test code_n(c) == 56 && code_k(c) == 8
+        A = reshape([1 + x^7], (1, 1))
+        B = reshape([1 + x^7 + s + x^8 + s*x^7 + x], (1, 1))
+        c = LPCode(A,B)
+        # [[56, 28, 2]] 2BGA code
+        @test code_n(c) == 56 && code_k(c) == 28
+    end
+end