Fix matrix rank issues and run all code analyses

- Fixed improved staircase: below-diagonal connections preserve full
  parity rank (col7->row0 s3, col1->row4 s15)
- Fixed PEG matrix: staircase backbone with cross-connections,
  all parity cols dv>=2, VN degrees [7,3,3,3,2,2,2,2]
- Clean up VN degree display (remove np.int64 wrapper)
- Ran all four analyses with 200 frames per point

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

data/analysis_results.json

@@ -0,0 +1,166 @@
+{
+  "matrix_compare": [
+    {
+      "matrix": "Original staircase",
+      "lam_s": 0.5,
+      "fer": 0.99,
+      "avg_iter": 29.81
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 0.5,
+      "fer": 1.0,
+      "avg_iter": 30.0
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 0.5,
+      "fer": 1.0,
+      "avg_iter": 30.0
+    },
+    {
+      "matrix": "Original staircase",
+      "lam_s": 1.0,
+      "fer": 1.0,
+      "avg_iter": 30.0
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 1.0,
+      "fer": 1.0,
+      "avg_iter": 30.0
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 1.0,
+      "fer": 1.0,
+      "avg_iter": 30.0
+    },
+    {
+      "matrix": "Original staircase",
+      "lam_s": 1.5,
+      "fer": 0.985,
+      "avg_iter": 29.565
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 1.5,
+      "fer": 1.0,
+      "avg_iter": 30.0
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 1.5,
+      "fer": 0.985,
+      "avg_iter": 29.565
+    },
+    {
+      "matrix": "Original staircase",
+      "lam_s": 2.0,
+      "fer": 0.81,
+      "avg_iter": 24.49
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 2.0,
+      "fer": 0.925,
+      "avg_iter": 27.825
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 2.0,
+      "fer": 0.955,
+      "avg_iter": 28.695
+    },
+    {
+      "matrix": "Original staircase",
+      "lam_s": 3.0,
+      "fer": 0.38,
+      "avg_iter": 12.02
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 3.0,
+      "fer": 0.41,
+      "avg_iter": 12.89
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 3.0,
+      "fer": 0.32,
+      "avg_iter": 10.28
+    },
+    {
+      "matrix": "Original staircase",
+      "lam_s": 4.0,
+      "fer": 0.105,
+      "avg_iter": 4.045
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 4.0,
+      "fer": 0.1,
+      "avg_iter": 3.9
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 4.0,
+      "fer": 0.07,
+      "avg_iter": 3.03
+    },
+    {
+      "matrix": "Original staircase",
+      "lam_s": 5.0,
+      "fer": 0.14,
+      "avg_iter": 5.06
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 5.0,
+      "fer": 0.04,
+      "avg_iter": 2.16
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 5.0,
+      "fer": 0.04,
+      "avg_iter": 2.16
+    },
+    {
+      "matrix": "Original staircase",
+      "lam_s": 7.0,
+      "fer": 0.015,
+      "avg_iter": 1.435
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 7.0,
+      "fer": 0.005,
+      "avg_iter": 1.145
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 7.0,
+      "fer": 0.005,
+      "avg_iter": 1.145
+    },
+    {
+      "matrix": "Original staircase",
+      "lam_s": 10.0,
+      "fer": 0.005,
+      "avg_iter": 1.145
+    },
+    {
+      "matrix": "Improved staircase",
+      "lam_s": 10.0,
+      "fer": 0.0,
+      "avg_iter": 1.0
+    },
+    {
+      "matrix": "PEG ring",
+      "lam_s": 10.0,
+      "fer": 0.0,
+      "avg_iter": 1.0
+    }
+  ]
+}

model/ldpc_analysis.py

@@ -401,7 +401,8 @@ def build_improved_staircase(z=32):
     connections to low-degree columns.
 
     Key: col 7 gets extra connection (dv=1->dv=2), col 1 gets extra (dv=2->dv=3).
-    NOTE: This is NO LONGER purely lower-triangular. Must use peg_encode.
+    Extra connections are BELOW the staircase diagonal to preserve
+    lower-triangular parity structure and full rank.
     """
     m_base = 7
     n_base = 8
@@ -410,12 +411,12 @@ def build_improved_staircase(z=32):
     H_b = H_BASE.copy()
 
     # Add extra connection for col 7 (currently only connected from row 6)
-    # Connect col 7 to row 3 with shift 17
+    # Row 0 is below col 7's diagonal (row 6) in the staircase sense
-    H_b[3, 7] = 17
+    H_b[0, 7] = 3
 
     # Add extra connection for col 1 (currently rows 0,1)
-    # Connect col 1 to row 4 with shift 11
+    # Row 4 is below col 1's diagonal (row 0)
-    H_b[4, 1] = 11
+    H_b[4, 1] = 15
 
     # Build full matrix
     m_full = m_base * z
@@ -435,51 +436,37 @@ def build_improved_staircase(z=32):
 
 def build_peg_matrix(z=32):
     """
-    PEG-like construction with ring structure (not strictly lower-triangular).
+    PEG-like construction with more uniform degree distribution.
 
-    More uniform degree distribution. All VN columns have degree >= 2.
+    Uses staircase parity backbone (guaranteeing full parity rank and
+    sequential encoding) plus below-diagonal cross-connections for
+    better girth and degree distribution.
+
+    VN degrees: col0=7, cols1-3=3, cols4-7=2 (avg 2.75 vs 2.5 original).
+    All parity columns have dv >= 2 (no weak degree-1 nodes).
     """
     m_base = 7
     n_base = 8
-
-    # PEG ring: each column connects to 2-3 rows, more evenly distributed
-    # Design for good girth while maintaining encodability
     H_b = -np.ones((m_base, n_base), dtype=np.int16)
 
-    # Column 0 (info): high degree, connect to rows 0,1,2,3,4,5,6
+    # Column 0 (info): connect to all 7 rows with spread shifts
-    for r in range(m_base):
+    H_b[0, 0] = 0;  H_b[1, 0] = 7;   H_b[2, 0] = 13
-        H_b[r, 0] = (r * z // m_base) % z
+    H_b[3, 0] = 19; H_b[4, 0] = 25;  H_b[5, 0] = 3; H_b[6, 0] = 9
 
-    # Parity columns with ring connections (degree 2-3 each)
+    # Staircase parity backbone (lower-triangular, full rank guaranteed)
-    # Col 1: rows 0, 1, 5
     H_b[0, 1] = 5
-    H_b[1, 1] = 3
+    H_b[1, 1] = 3;  H_b[1, 2] = 0
-    H_b[5, 1] = 19
+    H_b[2, 2] = 11; H_b[2, 3] = 0
+    H_b[3, 3] = 17; H_b[3, 4] = 0
+    H_b[4, 4] = 23; H_b[4, 5] = 0
+    H_b[5, 5] = 29; H_b[5, 6] = 0
+    H_b[6, 6] = 5;  H_b[6, 7] = 0
 
-    # Col 2: rows 1, 2, 6
+    # Below-diagonal cross-connections for better degree distribution
-    H_b[1, 2] = 0
+    H_b[3, 1] = 21  # col 1 dv=2->3
-    H_b[2, 2] = 7
+    H_b[5, 2] = 15  # col 2 dv=2->3
-    H_b[6, 2] = 23
+    H_b[6, 3] = 27  # col 3 dv=2->3
-
+    H_b[0, 7] = 0   # col 7 dv=1->2
-    # Col 3: rows 2, 3
-    H_b[2, 3] = 0
-    H_b[3, 3] = 13
-
-    # Col 4: rows 3, 4
-    H_b[3, 4] = 0
-    H_b[4, 4] = 19
-
-    # Col 5: rows 4, 5
-    H_b[4, 5] = 0
-    H_b[5, 5] = 25
-
-    # Col 6: rows 5, 6
-    H_b[5, 6] = 0
-    H_b[6, 6] = 31
-
-    # Col 7: rows 0, 6
-    H_b[0, 7] = 11
-    H_b[6, 7] = 0
 
     # Build full matrix
     m_full = m_base * z
@@ -708,7 +695,7 @@ def run_matrix_comparison(lam_b=0.1, n_frames=500, max_iter=30):
             vn_degrees.append(d)
         girth = compute_girth(H_b, 32)
         girth_str = str(girth) if girth < float('inf') else 'inf'
-        deg_str = str(vn_degrees)
+        deg_str = str([int(d) for d in vn_degrees])
         print(f"  {name:<22s} {deg_str:<30s} {girth_str:>6s}")
 
     # Run simulations