Channel Codes

Reed-Solomon Codes

Reed-Solomon codes are non-binary codes with block length of N=2m-1 symbols, that can correct t symbol errors, if a hard decision decoder is used (for sophisticated soft decision decoding see our publications here). K is given by K=N-2t. Reed-Solomon codes are MDS codes and therefore achieve the maximum possible minimum distance (in a non-binary sense). Thus the code structure itself is immune against the error floor problematic of turbo and LDPC codes.

Some Reed-Solomon codes are of special interest and have been adopted in many communication standards, in some cases also as shortened version (list is not exhaustive):

CodeApplication / Standard
RS(255,239)G.fast, optical, DVB, DRM
RS(255,223)space comm (NASA, ESA)
RS(255,245)DAB+
RS(255,x)WiMAX, DSL
RS(63,55)WPAN (802.15.4), DVB
RS(15,x)visible light comm.

The parity check matrix can be described with two different structures: the vandermonde form and the strip (cyclic) form.

The binary image representation provides an equivalent binary code, frequently used for code analysis and soft decision decoding. The binary image transformation is the same as for non-binary LDPC codes described here.

N=7

N K Rate t non-binary matrix Nb Kb binary image
7 5 0.71 1 strip, vanderm 21 15 strip, vanderm, 1-min
7 3 0.43 2 strip, vanderm 21 9 strip, vanderm, 1-min


N=15

N K Rate t non-binary matrix Nb Kb binary image
15 13 0.87 1 strip, vanderm 60 52 strip, vanderm, 1-min
15 11 0.73 2 strip, vanderm 60 44 strip, vanderm, 1-min
15 9 0.60 3 strip, vanderm 60 36 strip, vanderm, 1-min
15 7 0.47 4 strip, vanderm 60 28 strip, vanderm, 1-min
15 5 0.33 5 strip, vanderm 60 20 strip, vanderm, 1-min
15 3 0.20 6 strip, vanderm 60 12 strip, vanderm, 1-min


N=31

N K Rate t non-binary matrix Nb Kb binary image
31 29 0.94 1 strip, vanderm 155 145 strip, vanderm, 1-min
31 27 0.87 2 strip, vanderm 155 135 strip, vanderm, 1-min
31 25 0.81 3 strip, vanderm 155 125 strip, vanderm, 1-min
31 23 0.74 4 strip, vanderm 155 115 strip, vanderm, 1-min
31 21 0.68 5 strip, vanderm 155 105 strip, vanderm, 1-min
31 19 0.61 6 strip, vanderm 155 95 strip, vanderm, 1-min
31 17 0.55 7 strip, vanderm 155 85 strip, vanderm, 1-min
31 15 0.48 8 strip, vanderm 155 75 strip, vanderm, 1-min
31 13 0.42 9 strip, vanderm 155 65 strip, vanderm, 1-min
31 11 0.35 10 strip, vanderm 155 55 strip, vanderm, 1-min
31 9 0.29 11 strip, vanderm 155 45 strip, vanderm, 1-min
31 7 0.23 12 strip, vanderm 155 35 strip, vanderm, 1-min
31 5 0.16 13 strip, vanderm 155 25 strip, vanderm, 1-min
31 3 0.10 14 strip, vanderm 155 15 strip, vanderm, 1-min


N=63

N K Rate t non-binary matrix Nb Kb binary image
63 61 0.97 1 strip, vanderm 378 366 strip, vanderm
63 59 0.94 2 strip, vanderm 378 354 strip, vanderm
63 57 0.90 3 strip, vanderm 378 342 strip, vanderm
63 55 0.87 4 strip, vanderm 378 330 strip, vanderm, 1-min
63 53 0.84 5 strip, vanderm 378 318 strip, vanderm
63 51 0.81 6 strip, vanderm 378 306 strip, vanderm
63 49 0.78 7 strip, vanderm 378 294 strip, vanderm
63 47 0.75 8 strip, vanderm 378 282 strip, vanderm
63 45 0.71 9 strip, vanderm 378 270 strip, vanderm
63 43 0.68 10 strip, vanderm 378 258 strip, vanderm
63 41 0.65 11 strip, vanderm 378 246 strip, vanderm
63 39 0.62 12 strip, vanderm 378 234 strip, vanderm
63 37 0.59 13 strip, vanderm 378 222 strip, vanderm
63 35 0.56 14 strip, vanderm 378 210 strip, vanderm
63 33 0.52 15 strip, vanderm 378 198 strip, vanderm, 1-min
63 31 0.49 16 strip, vanderm 378 186 strip, vanderm
63 29 0.46 17 strip, vanderm 378 174 strip, vanderm
63 27 0.43 18 strip, vanderm 378 162 strip, vanderm
63 25 0.40 19 strip, vanderm 378 150 strip, vanderm
63 23 0.37 20 strip, vanderm 378 138 strip, vanderm
63 21 0.33 21 strip, vanderm 378 126 strip, vanderm
63 19 0.30 22 strip, vanderm 378 114 strip, vanderm
63 17 0.27 23 strip, vanderm 378 102 strip, vanderm
63 15 0.24 24 strip, vanderm 378 90 strip, vanderm
63 13 0.21 25 strip, vanderm 378 78 strip, vanderm
63 11 0.17 26 strip, vanderm 378 66 strip, vanderm
63 9 0.14 27 strip, vanderm 378 54 strip, vanderm
63 7 0.11 28 strip, vanderm 378 42 strip, vanderm

 

 

N=127

N K Rate t non-binary matrix Nb Kb binary image
127 125 0.98 1 strip, vanderm 889 875 strip, vanderm
127 123 0.97 2 strip, vanderm 889 861 strip, vanderm
127 121 0.95 3 strip, vanderm 889 847 strip, vanderm
127 119 0.94 4 strip, vanderm 889 833 strip, vanderm
127 117 0.92 5 strip, vanderm 889 819 strip, vanderm
127 115 0.91 6 strip, vanderm 889 805 strip, vanderm
127 113 0.89 7 strip, vanderm 889 791 strip, vanderm
127 111 0.87 8 strip, vanderm 889 777 strip, vanderm, 1-min
127 109 0.86 9 strip, vanderm 889 763 strip, vanderm
127 107 0.84 10 strip, vanderm 889 749 strip, vanderm
127 105 0.83 11 strip, vanderm 889 735 strip, vanderm
127 103 0.81 12 strip, vanderm 889 721 strip, vanderm
127 101 0.80 13 strip, vanderm 889 707 strip, vanderm
127 99 0.78 14 strip, vanderm 889 693 strip, vanderm
127 97 0.76 15 strip, vanderm 889 679 strip, vanderm
127 95 0.75 16 strip, vanderm 889 665 strip, vanderm
127 93 0.73 17 strip, vanderm 889 651 strip, vanderm
127 91 0.72 18 strip, vanderm 889 637 strip, vanderm
127 89 0.70 19 strip, vanderm 889 623 strip, vanderm
127 87 0.69 20 strip, vanderm 889 609 strip, vanderm
127 85 0.67 21 strip, vanderm 889 595 strip, vanderm
127 83 0.65 22 strip, vanderm 889 581 strip, vanderm
127 81 0.64 23 strip, vanderm 889 567 strip, vanderm
127 79 0.62 24 strip, vanderm 889 553 strip, vanderm
127 77 0.61 25 strip, vanderm 889 539 strip, vanderm
127 75 0.59 26 strip, vanderm 889 525 strip, vanderm
127 73 0.57 27 strip, vanderm 889 511 strip, vanderm
127 71 0.56 28 strip, vanderm 889 497 strip, vanderm
127 69 0.54 29 strip, vanderm 889 483 strip, vanderm
127 67 0.53 30 strip, vanderm 889 469 strip, vanderm, 1-min
127 65 0.51 31 strip, vanderm 889 455 strip, vanderm
127 63 0.50 32 strip, vanderm 889 441 strip, vanderm
127 61 0.48 33 strip, vanderm 889 427 strip, vanderm
127 59 0.46 34 strip, vanderm 889 413 strip, vanderm
127 57 0.45 35 strip, vanderm 889 399 strip, vanderm
127 55 0.43 36 strip, vanderm 889 385 strip, vanderm
127 53 0.42 37 strip, vanderm 889 371 strip, vanderm
127 51 0.40 38 strip, vanderm 889 357 strip, vanderm
127 49 0.39 39 strip, vanderm 889 343 strip, vanderm
127 47 0.37 40 strip, vanderm 889 329 strip, vanderm
127 45 0.35 41 strip, vanderm 889 315 strip, vanderm
127 43 0.34 42 strip, vanderm 889 301 strip, vanderm
127 41 0.32 43 strip, vanderm 889 287 strip, vanderm
127 39 0.31 44 strip, vanderm 889 273 strip, vanderm
127 37 0.29 45 strip, vanderm 889 259 strip, vanderm
127 35 0.28 46 strip, vanderm 889 245 strip, vanderm
127 33 0.26 47 strip, vanderm 889 231 strip, vanderm
127 31 0.24 48 strip, vanderm 889 217 strip, vanderm
127 29 0.23 49 strip, vanderm 889 203 strip, vanderm
127 27 0.21 50 strip, vanderm 889 189 strip, vanderm
127 25 0.20 51 strip, vanderm 889 175 strip, vanderm
127 23 0.18 52 strip, vanderm 889 161 strip, vanderm
127 21 0.17 53 strip, vanderm 889 147 strip, vanderm
127 19 0.15 54 strip, vanderm 889 133 strip, vanderm
127 17 0.13 55 strip, vanderm 889 119 strip, vanderm
127 15 0.12 56 strip, vanderm 889 105 strip, vanderm
127 13 0.10 57 strip, vanderm 889 91 strip, vanderm

 

 

N=255

N K Rate t non-binary matrix Nb Kb binary image
255 253 0.99 1 strip, vanderm 2040 2024 strip, vanderm
255 251 0.98 2 strip, vanderm 2040 2008 strip, vanderm
255 249 0.98 3 strip, vanderm 2040 1992 strip, vanderm
255 247 0.97 4 strip, vanderm 2040 1976 strip, vanderm
255 245 0.96 5 strip, vanderm 2040 1960 strip, vanderm
255 243 0.95 6 strip, vanderm 2040 1944 strip, vanderm
255 241 0.95 7 strip, vanderm 2040 1928 strip, vanderm
255 239 0.94 8 strip, vanderm 2040 1912 strip, vanderm, 1-min
255 237 0.93 9 strip, vanderm 2040 1896 strip, vanderm
255 235 0.92 10 strip, vanderm 2040 1880 strip, vanderm
255 233 0.91 11 strip, vanderm 2040 1864 strip, vanderm
255 231 0.91 12 strip, vanderm 2040 1848 strip, vanderm
255 229 0.90 13 strip, vanderm 2040 1832 strip, vanderm
255 227 0.89 14 strip, vanderm 2040 1816 strip, vanderm
255 225 0.88 15 strip, vanderm 2040 1800 strip, vanderm
255 223 0.87 16 strip, vanderm 2040 1784 strip, vanderm
255 221 0.87 17 strip, vanderm 2040 1768 strip, vanderm
255 219 0.86 18 strip, vanderm 2040 1752 strip, vanderm
255 217 0.85 19 strip, vanderm 2040 1736 strip, vanderm
255 215 0.84 20 strip, vanderm 2040 1720 strip, vanderm
255 213 0.84 21 strip, vanderm 2040 1704 strip, vanderm
255 211 0.83 22 strip, vanderm 2040 1688 strip, vanderm
255 209 0.82 23 strip, vanderm 2040 1672 strip, vanderm
255 207 0.81 24 strip, vanderm 2040 1656 strip, vanderm
255 205 0.80 25 strip, vanderm 2040 1640 strip, vanderm
255 203 0.80 26 strip, vanderm 2040 1624 strip, vanderm
255 201 0.79 27 strip, vanderm 2040 1608 strip, vanderm
255 199 0.78 28 strip, vanderm 2040 1592 strip, vanderm
255 197 0.77 29 strip, vanderm 2040 1576 strip, vanderm
255 195 0.76 30 strip, vanderm 2040 1560 strip, vanderm
255 193 0.76 31 strip, vanderm 2040 1544 strip, vanderm
255 191 0.75 32 strip, vanderm 2040 1528 strip, vanderm
255 189 0.74 33 strip, vanderm 2040 1512 strip, vanderm
255 187 0.73 34 strip, vanderm 2040 1496 strip, vanderm
255 185 0.73 35 strip, vanderm 2040 1480 strip, vanderm
255 183 0.72 36 strip, vanderm 2040 1464 strip, vanderm
255 181 0.71 37 strip, vanderm 2040 1448 strip, vanderm
255 179 0.70 38 strip, vanderm 2040 1432 strip, vanderm
255 177 0.69 39 strip, vanderm 2040 1416 strip, vanderm
255 175 0.69 40 strip, vanderm 2040 1400 strip, vanderm
255 173 0.68 41 strip, vanderm 2040 1384 strip, vanderm
255 171 0.67 42 strip, vanderm 2040 1368 strip, vanderm
255 169 0.66 43 strip, vanderm 2040 1352 strip, vanderm
255 167 0.65 44 strip, vanderm 2040 1336 strip, vanderm
255 165 0.65 45 strip, vanderm 2040 1320 strip, vanderm
255 163 0.64 46 strip, vanderm 2040 1304 strip, vanderm
255 161 0.63 47 strip, vanderm 2040 1288 strip, vanderm
255 159 0.62 48 strip, vanderm 2040 1272 strip, vanderm
255 157 0.62 49 strip, vanderm 2040 1256 strip, vanderm
255 155 0.61 50 strip, vanderm 2040 1240 strip, vanderm
255 153 0.60 51 strip, vanderm 2040 1224 strip, vanderm
255 151 0.59 52 strip, vanderm 2040 1208 strip, vanderm
255 149 0.58 53 strip, vanderm 2040 1192 strip, vanderm
255 147 0.58 54 strip, vanderm 2040 1176 strip, vanderm
255 145 0.57 55 strip, vanderm 2040 1160 strip, vanderm
255 143 0.56 56 strip, vanderm 2040 1144 strip, vanderm
255 141 0.55 57 strip, vanderm 2040 1128 strip, vanderm
255 139 0.55 58 strip, vanderm 2040 1112 strip, vanderm
255 137 0.54 59 strip, vanderm 2040 1096 strip, vanderm
255 135 0.53 60 strip, vanderm 2040 1080 strip, vanderm
255 133 0.52 61 strip, vanderm 2040 1064 strip, vanderm
255 131 0.51 62 strip, vanderm 2040 1048 strip, vanderm
255 129 0.51 63 strip, vanderm 2040 1032 strip, vanderm
255 127 0.50 64 strip, vanderm 2040 1016 strip, vanderm
255 119 0.47 68 strip, vanderm 2040 952 strip, vanderm
255 111 0.44 72 strip, vanderm 2040 888 strip, vanderm
255 103 0.40 76 strip, vanderm 2040 824 strip, vanderm
255 95 0.37 80 strip, vanderm 2040 760 strip, vanderm
255 87 0.34 84 strip, vanderm 2040 696 strip, vanderm
255 79 0.31 88 strip, vanderm 2040 632 strip, vanderm
255 71 0.28 92 strip, vanderm 2040 568 strip, vanderm
255 63 0.25 96 strip, vanderm 2040 504 strip, vanderm
255 55 0.22 100 strip, vanderm 2040 440 strip, vanderm
255 51 0.20 102 strip, vanderm 2040 408 strip, vanderm
255 47 0.18 104 strip, vanderm 2040 376 strip, vanderm
255 39 0.15 108 strip, vanderm 2040 312 strip, vanderm
255 31 0.12 112 strip, vanderm 2040 248 strip, vanderm
255 23 0.09 116 strip, vanderm 2040 184 strip, vanderm


N=511

N K Rate t non-binary matrix Nb Kb binary image
511 507 0.99 2 strip or vanderm 4599 4563 strip or vanderm
511 503 0.98 4 strip or vanderm 4599 4527 strip or vanderm
511 499 0.98 6 strip or vanderm 4599 4491 strip or vanderm
511 495 0.97 8 strip or vanderm 4599 4455 strip or vanderm
511 491 0.96 10 strip or vanderm 4599 4419 strip or vanderm
511 487 0.95 12 strip or vanderm 4599 4383 strip or vanderm
511 483 0.95 14 strip or vanderm 4599 4347 strip or vanderm
511 479 0.94 16 strip or vanderm 4599 4311 strip or vanderm
511 475 0.93 18 strip or vanderm 4599 4275 strip or vanderm
511 471 0.92 20 strip or vanderm 4599 4239 strip or vanderm
511 467 0.91 22 strip or vanderm 4599 4203 strip or vanderm
511 463 0.91 24 strip or vanderm 4599 4167 strip or vanderm
511 459 0.90 26 strip or vanderm 4599 4131 strip or vanderm
511 455 0.89 28 strip or vanderm 4599 4095 strip or vanderm
511 451 0.88 30 strip or vanderm 4599 4059 strip or vanderm
511 447 0.87 32 strip or vanderm 4599 4023 strip or vanderm
511 443 0.87 34 strip or vanderm 4599 3987 strip or vanderm
511 439 0.86 36 strip or vanderm 4599 3951 strip or vanderm
511 435 0.85 38 strip or vanderm 4599 3915 strip or vanderm
511 431 0.84 40 strip or vanderm 4599 3879 strip or vanderm
511 427 0.84 42 strip or vanderm 4599 3843 strip or vanderm
511 423 0.83 44 strip or vanderm 4599 3807 strip or vanderm
511 419 0.82 46 strip or vanderm 4599 3771 strip or vanderm
511 415 0.81 48 strip or vanderm 4599 3735 strip or vanderm
511 411 0.80 50 strip or vanderm 4599 3699 strip or vanderm
511 407 0.80 52 strip or vanderm 4599 3663 strip or vanderm
511 403 0.79 54 strip or vanderm 4599 3627 strip or vanderm
511 399 0.78 56 strip or vanderm 4599 3591 strip or vanderm
511 395 0.77 58 strip or vanderm 4599 3555 strip or vanderm
511 391 0.77 60 strip or vanderm 4599 3519 strip or vanderm
511 387 0.76 62 strip or vanderm 4599 3483 strip or vanderm
511 383 0.75 64 strip or vanderm 4599 3447 strip or vanderm
511 379 0.74 66 strip or vanderm 4599 3411 strip or vanderm
511 375 0.73 68 strip or vanderm 4599 3375 strip or vanderm
511 371 0.73 70 strip or vanderm 4599 3339 strip or vanderm
511 367 0.72 72 strip or vanderm 4599 3303 strip or vanderm
511 363 0.71 74 strip or vanderm 4599 3267 strip or vanderm
511 359 0.70 76 strip or vanderm 4599 3231 strip or vanderm
511 355 0.69 78 strip or vanderm 4599 3195 strip or vanderm
511 351 0.69 80 strip or vanderm 4599 3159 strip or vanderm
511 347 0.68 82 strip or vanderm 4599 3123 strip or vanderm
511 343 0.67 84 strip or vanderm 4599 3087 strip or vanderm
511 255 0.50 128 strip or vanderm 4599 2295 strip or vanderm


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