classical Sipser-Spielman expander codes

src/QuantumExpanders.jl

@@ -12,11 +12,12 @@ using Graphs
 using Oscar:coefficients
 using Graphs:add_edge!,nv,ne,neighbors,Graphs
 
+include("classical_expander_code.jl")
 include("morgenstern.jl")
 include("cayley_graphs.jl")
 include("tensor_codes.jl")
 
-export gen_code, gen_good_code, tanner_code, tanner_code_quadripartite, cayley_complex_square_graphs, cayley_complex_square_graphs_quadripartite, morgenstern_generators, alternative_morgenstern_generators
+export gen_code, gen_good_code, tanner_code, tanner_code_quadripartite, cayley_complex_square_graphs, cayley_complex_square_graphs_quadripartite, morgenstern_generators, alternative_morgenstern_generators, expander_code_parity_matrix
 
 ##
 

src/classical_expander_code.jl

@@ -0,0 +1,74 @@
+"""Generate a random binary parity check matrix with specified dimensions and rank properties.
+"""
+function random_parity_check_matrix(rows::Int, cols::Int; min_rank::Int=1, max_attempts::Int=100)
+    if rows <= 0 || cols <= 0
+        error("Matrix dimensions must be positive")
+    end
+    if min_rank > min(rows, cols)
+        error("Requested rank exceeds possible maximum")
+    end
+    for _ in 1:max_attempts
+        H = zeros(Int, rows, cols)
+        for c in 1:cols
+            while true
+                col = rand(0:1, rows)
+                sum(col) > 0 && (H[:,c] = col; break)
+            end
+        end
+        rk = rank(Matrix(H))
+        rk >= min_rank && return H
+    end
+end
+
+"""
+    expander_code_parity_matrix
+
+Construct the parity-check matrix for a classical expander code (a type of Tanner code).
+
+An expander code is defined by:
+- A **regular base graph** `g` (typically a good expander graph like a Ramanujan graph)
+- A **short inner code** specified by its parity-check matrix `H`
+
+The resulting code has:
+- Variables corresponding to edges of `g`
+- Constraints enforcing the inner code at each vertex of `g`
+
+The construction follows Sipser-Spielman expander codes:
+- Each edge in `g` becomes a variable in the code
+- Each vertex enforces its incident edges to satisfy the inner code
+- The code has rate ≥ 1 - (1 - r) * (n_edges/n_vertices) where r is the rate of the inner code.
+"""
+function expander_code_parity_matrix(g::AbstractGraph, H_inner::AbstractMatrix{<:Integer})
+    H_inner = sparse(H_inner)
+    n_vertices = nv(g)
+    n_edges = ne(g)
+    n_vertices == 0 && error("Graph must have vertices")
+    degrees = degree(g)
+    graph_degree = first(degrees)
+    all(==(graph_degree), degrees) || error("Graph must be regular")
+    _, inner_code_length = size(H_inner)
+    inner_code_length == graph_degree || error("Inner code length must match graph degree")
+    edge_list = collect(edges(g))
+    edge_indices = Dict{Edge,Int}()
+    for (idx, e) in enumerate(edge_list)
+        edge_indices[e] = idx
+    end
+    nnz_est = n_vertices * nnz(H_inner)
+    I = sizehint!(Int[], nnz_est)
+    J = sizehint!(Int[], nnz_est)
+    V = sizehint!(Int[], nnz_est)
+    for v in 1:n_vertices
+        incident_edges = [e for e in edge_list if src(e) == v || dst(e) == v]
+        sort!(incident_edges, by=e->(min(src(e),dst(e)), max(src(e),dst(e))))
+        for (check_row, row_vals) in enumerate(eachrow(H_inner))
+            for (edge_pos, val) in enumerate(row_vals)
+                val == 0 && continue
+                edge_idx = edge_indices[incident_edges[edge_pos]]
+                push!(I, (v-1)*size(H_inner,1) + check_row)
+                push!(J, edge_idx)
+                push!(V, val)
+            end
+        end
+    end
+    sparse(I, J, V, n_vertices*size(H_inner,1), n_edges)
+end

test/test_ramanujan.jl

@@ -0,0 +1,51 @@
+@testitem "Test classical  expander graph properties" begin
+    using Oscar
+    using LinearAlgebra
+    using Graphs
+    using Graphs: nv, neighbors, AbstractGraph, degree, ne
+    using QuantumExpanders
+    using LogExpFunctions
+
+    function generate_parity_matrix(rows::Int, cols::Int; min_rank::Int=1, max_attempts::Int=100)
+        if rows <= 0 || cols <= 0
+            error("Matrix dimensions must be positive")
+        end
+        if min_rank > min(rows, cols)
+            error("Requested rank exceeds possible maximum")
+        end
+        for _ in 1:max_attempts
+            H = zeros(Int, rows, cols)
+            for c in 1:cols
+                while true
+                    col = rand(0:1, rows)
+                    sum(col) > 0 && (H[:,c] = col; break)
+                end
+            end
+            rk = rank(Matrix(H))
+            rk >= min_rank && return H
+        end
+    end
+
+    @testset "Classical Expander code using Ramanujan Graphs" begin
+        test_pairs = [(5, 29),
+                      (13, 17),
+                      (29, 13)]
+        for (p, q) in test_pairs
+            g = ramanujan_graph(p, q)
+            H_inner = generate_parity_matrix(2, degree(g, 1))
+            @test size(H_inner, 2) == degree(g, 1)
+            H_expander = QuantumExpanders.expander_code_parity_matrix(g, H_inner)
+            m, n = size(H_expander)
+            r = rank(H_expander)
+            k = n - r
+            @test n == ne(g)
+            @test m == nv(g) * size(H_inner, 1)
+            @test k == n - r
+            d = degree(g,1)
+            inner_rate = (d - rank((H_inner))) / d
+            theoretical_min_rate = 2*inner_rate - 1
+            actual_rate = k/n
+            @test actual_rate ≥ theoretical_min_rate
+        end
+    end
+end