Haah's cubic code via LP construction method

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

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -331,3 +331,34 @@ function haah_cubic_codes(a_shifts::Array{Int}, b_shifts::Array{Int}, l::Int)
     b = sum(GA[n%dim(GA)+1] for n in b_shifts)
     two_block_group_algebra_codes(a, b)
 end
+
+"""
+Haah’s cubic code is defined as \$\\text{LP}(1 + x + y + z, 1 + xy + xz + yz)\$
+where \$\\text{LP}\$ is the lifted product code, and `x`, `y`, `z` are elements
+of the ring \$R = \\mathbb{F}_2[x, y, z] / (x^L - 1, y^L - 1, z^L - 1)\$. Here
+\$\\mathbb{F}_2\$ is the finite field of order `2` and `L` is the lattice size.
+The ring \$R\$ is the group algebra \$\\mathbb{F}_qG\$ of a finite group `G`, where
+\$G = (C_L)^3\$ and \$C_L\$ is the cyclic group of order `L`. This method of Haah's
+cubic code construction is outlined in Appendix B of [panteleev2022asymptotically](@cite).
+
+Here is an example of a [[1024, 30, 13 ≤ d ≤ 32]] Haah's cubic code from Appendix B,
+code D of [panteleev2021degenerate](@cite) on the `8 × 8 × 8` Lattice.
+
+```jldoctest
+julia> l = 8;
+
+julia> c = haah_cubic_codes(l);
+
+julia> code_n(c), code_k(c)
+(1024, 30)
+```
+
+See also: [`bicycle_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`two_block_group_algebra_codes`](@ref).
+"""
+function haah_cubic_codes(l::Int)
+    GA = group_algebra(GF(2), abelian_group([l,l,l]))
+    x, y, z = gens(GA)
+    c = [1 + x   + y   + z;;]
+    d = [1 + x*y + x*z + y*z;;]
+    LPCode(c,d)
+end