Maximum Likelihood (ML) Simulation Results
Here, maximum likelihood decoding (ML) simulation results for various channel codes can be downloaded, including:
- standardized LDPC (WiMAX, WRAN etc.)
- LDPC of special interest (MacKay, Tanner etc.)
- non-binary LDPC
- Array LDPC
- LTE Turbo
- BCH
- Reed-Solomon
- Polar Codes
- Reed-Muller Codes
For some of the codes, we furthermore provide simulation results with LP decoding (LP) and improved LP decoding with adaptive cut generation (LP+ACG). Our software toolbox containing algorithms for ML and LP decoding is available here.
The results on this website are extended continuously.
We gratefully thank the AHRP cluster on which large parts of the computations were performed.
How to Cite
If you use these results in your publication, please cite our website as:
Michael Helmling, Stefan Scholl, Florian Gensheimer, Tobias Dietz, Kira Kraft, Stefan Ruzika, and Norbert Wehn. Database of Channel Codes and ML Simulation Results. www.rptu.de/channel-codes, 2023.
LDPC Codes
| Name | N | K | rate | dmin | IP/ML | Code |
|---|---|---|---|---|---|---|
| WiMAX (802.16) | 576 | 480 | 5/6 | 5 | sim results | alist |
WiMAX (802.16) | 672 | 560 | 5/6 | 7 | sim results | alist |
| WiMAX (802.16) | 768 | 640 | 5/6 | 7 | sim results | alist |
| WiMAX (802.16) | 864 | 720 | 5/6 | 7 | sim results | alist |
| WiMAX (802.16) | 960 | 800 | 5/6 | 7 | sim results | alist |
| WiMAX (802.16) | 1056 | 880 | 5/6 | 7 | sim results | alist |
| LDPC (TU KL) | 96 | 48 | 1/2 | 10 | sim results | alist |
| Wimax-like | 192 | 92 | 1/2 | 10 | sim results | alist |
| Wimax-like | 240 | 120 | 1/2 | 11 | sim results | alist |
| Wimax-like | 336 | 168 | 1/2 | 15 | sim results | alist |
| Wimax-like | 384 | 192 | 1/2 | 15 | sim results | alist |
| WRAN (802.22) | 384 | 192 | 1/2 | 15 | alist | |
| WRAN (802.22) | 384 | 256 | 2/3 | 8 | sim results | alist |
| WRAN (802.22) | 384 | 288 | 3/4 | 8 | sim results | alist |
| WRAN (802.22) | 384 | 320 | 5/6 | 4 | sim results | alist |
| WRAN (802.22) | 480 | 240 | 1/2 | 16 | alist | |
| WRAN (802.22) | 480 | 320 | 2/3 | 12 | alist | |
| WRAN (802.22) | 480 | 360 | 3/4 | 7 | sim results | alist |
| WRAN (802.22) | 480 | 400 | 5/6 | 6 | sim results | alist |
| WiFi (802.11) | 648 | 540 | 5/6 | 8 | sim results | alist |
| ITU G.h | 336 | 168 | 1/2 | 13 | alist | |
| CCSDS | 32 | 16 | 1/2 | 4 | sim results | alist |
| CCSDS | 128 | 64 | 1/2 | 14 | sim results | alist |
| Tanner (3,5) | 155 | 64 | ≈0.413 | 20 | sim results | alist |
| 96 | 48 | 1/2 | 6 | sim results | alist | |
| MacKay 204.33.484 | 204 | 102 | 1/2 | 8 | sim results | alist |
MacKay 408.33.844 | 408 | 204 | 1/2 | 14 | alist | |
| Multi-Edge Type | 100 | 50 | 1/2 | 10 | sim results | alist |
| Multi-Edge Type v0 | 120 | 60 | 1/2 | 10 | sim results | alist |
| Multi-Edge Type v1 | 120 | 60 | 1/2 | 10 | sim results | alist |
| Multi-Edge Type | 128 | 64 | 1/2 | sim results | alist | |
| Multi-Edge Type | 200 | 100 | 1/2 | 12 | alist | |
| Multi-Edge Type | 840 | 420 | 1/2 | 18 | alist |
FER charts LDPC
Non-binary LDPC Codes
ML simulation results have been obtained via the binary image representation of the non binary codes. The alists and matrices for the non-binary codes can be found in the non-binary LDPC database.
| Name | Nbin | Kbin | GF | rate | dmin | IP/ML | Code |
|---|---|---|---|---|---|---|---|
| TU KL Code | 96 | 48 | 64 | 1/2 | 11 | ML results | here |
TU KL Code | 128 | 64 | 256 | 1/2 | 13 | ML results | here |
| TU KL Code | 128 | 64 | 16 | 1/2 | 9 | ML results | here |
| U-NBPB (CCSDS) | 128 | 64 | 256 | 1/2 | 14 | ML results | here |
| TU KL Code | 576 | 480 | 64 | 5/6 | 6 | ML results | here |
| TU KL Code | 576 | 480 | 256 | 5/6 | 7 | here |
FER charts Non-Binary LDPC
Array LDPC Codes
| Name | N | K | p | r | rate | dmin | IP/ML | Code |
|---|---|---|---|---|---|---|---|---|
| Array | 49 | 24 | 7 | 4 | 0.49 | 8 | ML results | alist |
Array | 49 | 30 | 7 | 3 | 0.61 | 6 | ML results | alist |
| Array | 49 | 36 | 7 | 2 | 0.73 | 4 | ML results | alist |
| Array | 49 | 42 | 7 | 1 | 0.86 | 2 | ML results | alist |
| Array | 121 | 60 | 11 | 6 | 0.50 | 16 | ML results | alist |
| Array | 121 | 70 | 11 | 5 | 0.58 | 10 | ML results | alist |
| Array | 121 | 80 | 11 | 4 | 0.66 | 10 | ML results | alist |
| Array | 121 | 90 | 11 | 3 | 0.74 | 6 | ML results | alist |
| Array | 121 | 100 | 11 | 2 | 0.83 | 4 | ML results | alist |
| Array | 121 | 110 | 11 | 1 | 0.91 | 2 | ML results | alist |
| Array | 169 | 96 | 13 | 6 | 0.57 | 14 | alist | |
| Array | 169 | 108 | 13 | 5 | 0.64 | 12 | ML results | alist |
| Array | 169 | 120 | 13 | 4 | 0.71 | 10 | ML results | alist |
| Array | 169 | 132 | 13 | 3 | 0.78 | 6 | ML results | alist |
| Array | 169 | 144 | 13 | 2 | 0.85 | 4 | ML results | alist |
| Array | 169 | 156 | 13 | 1 | 0.92 | 2 | ML results | alist |
| Array | 361 | 288 | 19 | 4 | 0.80 | 16 | ML results | alist |
| Array | 361 | 324 | 19 | 2 | 0.90 | 4 | ML results | alist |
| Array | 529 | 440 | 23 | 4 | 0.83 | 10 | alist | |
| Array | 1369 | 1224 | 37 | 4 | 0.89 | 10 | alist | |
| Array | 1369 | 1296 | 37 | 2 | 0.95 | 4 | alist | |
| Array | 1369 | 1332 | 37 | 1 | 0.97 | 2 | ML results | alist |
Turbo Codes
| Name | N | K | rate | dmin | IP/ML | Code |
|---|---|---|---|---|---|---|
| LTE | 132 | 40 | 1/3 | 11 | sim results | alist.xz |
| LTE | 156 | 48 | 1/3 | 13 | sim results | alist.xz |
| LTE | 180 | 56 | 1/3 | 13 | sim results | alist.xz |
| LTE | 204 | 64 | 1/3 | sim results | alist.xz | |
| LTE | 228 | 72 | 1/3 | sim results | alist.xz | |
| LTE | 252 | 80 | 1/3 | sim results | alist.xz | |
| LTE | 276 | 88 | 1/3 | sim results | alist.xz | |
| LTE | 300 | 96 | 1/3 | sim results | alist.xz | |
| LTE | 324 | 104 | 1/3 | sim results | alist.xz | |
| LTE | 348 | 112 | 1/3 | sim results | alist.xz | |
| LTE | 372 | 120 | 1/3 | sim results | alist.xz | |
| LTE | 396 | 128 | 1/3 | sim results | alist.xz |
FER charts Turbo
BCH Codes
| Name | N | K | rate | t | IP/ML | Codes |
|---|---|---|---|---|---|---|
| BCH | 7 | 4 | 0.57 | 1 | sim results | |
| BCH | 15 | 7 | 0.47 | 2 | sim_results | alist |
| BCH | 15 | 11 | 0.73 | 1 | sim_results | alist |
| BCH | 31 | 11 | 0.35 | 5 | sim_results | alist |
| BCH | 31 | 16 | 0.52 | 3 | sim results | alist |
| BCH | 31 | 21 | 0.68 | 2 | sim_results | alist |
| BCH | 31 | 26 | 0.84 | 1 | sim_results | alist |
| BCH | 63 | 30 | 0.48 | 6 | sim_results | alist |
| BCH | 63 | 36 | 0.57 | 5 | sim_results | alist |
| BCH | 63 | 39 | 0.62 | 4 | sim_results | alist |
| BCH | 63 | 45 | 0.71 | 3 | sim_results | alist |
| BCH | 63 | 51 | 0.81 | 2 | sim_results | alist |
| BCH | 63 | 57 | 0.90 | 1 | sim_results | alist |
| BCH | 127 | 85 | 0.67 | 6 | sim results | alist |
| BCH | 127 | 92 | 0.72 | 5 | sim_results | alist |
| BCH | 127 | 99 | 0.78 | 4 | sim_results | alist |
| BCH | 127 | 106 | 0.83 | 3 | sim_results | alist |
| BCH | 127 | 113 | 0.89 | 2 | sim_results | alist |
| BCH | 127 | 120 | 0.94 | 1 | sim results | alist |
| BCH | 255 | 223 | 0.88 | 4 | sim results | alist |
| BCH | 255 | 231 | 0.91 | 3 | sim results | alist |
| BCH | 255 | 239 | 0.94 | 2 | sim results | alist |
| BCH | 255 | 247 | 0.97 | 1 | sim results | alist |
| BCH | 511 | 484 | 0.95 | 3 | sim results | alist |
| BCH | 511 | 493 | 0.96 | 2 | sim results | alist |
| BCH | 511 | 502 | 0.98 | 1 | sim results | alist |
| BCH | 1023 | 993 | 0.97 | 3 | sim results | alist |
| BCH | 1023 | 1003 | 0.98 | 2 | sim results | alist |
| BCH | 1023 | 1013 | 0.99 | 1 | sim results | alist |
FER charts BCH
Reed-Solomon Codes
| Name | N | K | rate | t | IP/ML | LP | Codes |
|---|---|---|---|---|---|---|---|
| RS | 7 | 3 | 0.43 | 2 | ML | binary image | |
| RS | 7 | 5 | 0.71 | 1 | ML | binary image | |
| RS | 15 | 3 | 0.20 | 6 | ML | binary image | |
| RS | 15 | 5 | 0.33 | 5 | ML | binary image | |
| RS | 15 | 7 | 0.47 | 4 | ML | binary image | |
| RS | 15 | 9 | 0.60 | 3 | ML | binary image | |
| RS | 15 | 11 | 0.73 | 2 | ML | binary image | |
| RS | 15 | 13 | 0.87 | 1 | ML | binary image | |
| RS | 31 | 23 | 0.74 | 4 | ML | binary image | |
| RS | 31 | 25 | 0.81 | 3 | ML | binary image | |
| RS | 31 | 27 | 0.87 | 2 | binary image | ||
| RS | 31 | 29 | 0.94 | 1 | ML | binary image | |
| RS | 63 | 55 | 0.87 | 4 | binary image |
FER charts Reed-Solomon
Polar Codes
Go to our polar codes page for more information about polar codes and their construction. The successive cancellation (SC) list decoding algorithm is described in
Tal, I. & Vardy, A: List decoding of polar codes. Proceedings of the IEEE International Symposium on Information Theory, St. Petersburg, Russia, Jul./Aug. 2011
FER charts Polar Codes
Other Codes
| Name | N | K | rate | dmin | IP/ML | Codes |
|---|---|---|---|---|---|---|
| ex. Golay | 24 | 12 | 1/2 | 8 | ML results | alist |
| cyclic code | 48 | 24 | 1/2 | 12 | ML results | alist |
Notes and References
Notes and references for the codes and decoding algorithms listed above.
LP Decoding
Feldman, J.; Wainwright, M. J. & Karger, D. R.:Using Linear Programming to Decode Binary Linear Codes. IEEE Transactions on Information Theory, vol. 51, 2005, pp. 954-972
Adaptive Cut Generation
Zhang, X. & Siegel, P. H.:Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes. IEEE Transactions on Information Theory, vol. 58, 2012, pp. 6581-6594
WRAN Codes (IEEE 802.22)
Codes taken from the WRAN (Wireless Regional Area Networks) standard; see http://www.ieee802.org/22/
MacKay Codes
The matrices and more information about codes by David J. C. MacKay can be found on his website:
Array LDPC Codes
Fan, John L.:Array codes as low-density parity-check codes. Proceedings of the 2nd International Symposium on Turbo Codes and Related Topics, Brest, France, 2000.