bussey1000

188 GALOIS FIELD TABLES. [Jan., TABLES OF GALOIS FIELDS OF ORDER LESS THAN 1,000. BY PROFESSOR W. II. BUSSE Y. (Read before the American Mathematical Society, September 13, 1909.) EVERY field of a finite number of marks may be represented as a Galois field of order s = pn} where pn is a power of a prime. The GF[pn~\ is defined uniquely by its order, and is therefore independent of the particular irreducible congruence used in its construction. In each of the following tables the GF[pn'] is constructed by means of a primitive irreducible congruence which appears at the top of the table. The marks of each field are arranged in two tables. In each table each mark appears as a power of a primitive root i, and also as a polynomial in i of degree h = n — 1. The coefficients in this polynomial are integers reduced modulo^. The mark Aik + J5^-I _| h Di + E, A 4= 0, is denoted by AB • • • DE, a symbol consisting of its detached coefficients in order. Zero coefficients must not be omitted. This is the usual symbol for a positive integer in the notation of the number system whose base is p. In the first table the marks are arranged according to ascending powers of i. In the second table the marks are arranged so that the symbols AB • • • DE represent the positive integers in natural order. By means of these two tables it is possible to perform with ease the operations of addition, sub­ traction, multiplication and division, within the field. The foregoing paragraph is reprinted from a paper entitled " Galois field tables for pn = 169," published in the BULLETIN, volume 12 (1905), pages 21-38. The present paper is an exten­ sion of that work and contains tables made according to the same plan but more compactly printed. The tables for fields of orders 28 and 29 are printed completely as were all of the tables in the paper mentioned. The other tables are abbreviated according to the following plan : The first of the two tables for a field of order pn contains the marks iK for which X = 1, 2, 3, • • -, v (where v is the smallest value of \ such that ix is equal to one of the numbers 1, 2, 3, • • -, p — 1) and also the marks ik for which X is a multiple of v. The table expresses each of these marks as a polynomial in i of degree k~n— 1. If X 1910.] GALOIS FIELD TABLES. 189 be any other exponent, the value of iK as a polynomial in i may be obtained by writing X = qv + r, where qv is the largest multiple of v less than X and consequently r is less than v. Then iK = iqv+r == iqv • ir. The values of iqv and ir may be read off directly from the table and their product obtained mentally because the value of iqv is integral. The second of the two tables for a field of order pn contains the integral marks 1, 2, 3^ ... y p _ 1 and also the marks aik + /3ik~l + yik~2 -f ... for which a = 1. It expresses each of these marks as a power iK of the primitive root i. If a 4= 1, the mark may be written a [> + ^*-i + ^-2 + .. 1 the divisions by a, modulo p, being made mentally or by means of the ordinary tables of indices for the prime p. The second table gives the value of a and the value of the ex­ pression in brackets each as a power of i. Their product is a power of i obtained by adding exponents. It is convenient to use detached coefficients and denote the mark aik + /3ik~l + yik~2 + • • • by the symbol afiy •. These coefficients are separated by commas when any one of them is an integer of more than one digit, that is when the modulus p is an integer of more than one digit. Otherwise it is more con­ venient not to make such a separation. This paper and the one published in 1905 contain tables of all Galois fields of order pn < 1000, n > 1. This is the limit set by Jacobi in his Canon Arithmeticus, which contains tables of indices for all primes less than 1000, i. e., tables of all Galois fields of prime order p < 1000. Example. The first table for the GF\1V~\ contains the marks iK for which X = 1, 2, 3, • • -, 18 or X = 18c. The table gives the value of each of these marks as a polynomial ai + y8. To express iU3 as a polynomial of the form ai + /3, write i243 = i2U. i9. From the table, i2M = 12 and i9 = 1U + 3. Conse­ quently t243 = 12(1 H + 3) = 132i + 36 = ISi + 2 because 132 S3 13 and 36 = 2 modulo 17. The second table contains in order the integral marks 1, 2, 3, •••, 16 and the marks ai 4- /3 for which a = 1. It gives the value of each of these marks as a power of i. The mark Si + 7 does not occur in the table because a = 8. To express it as a power of i, write Si + 7 = S(i -f |) = S(i + 3) because | = 3, modulo 17. From the table, 8 = i180 and i + 3 = i150. Therefore Si + 7 = im+im = im = i42 because i288 = 1. 190 GALOIS FIELD TABLES. [Jan., OF [35], i5 = i + 2, modulo 3. i* = ai4 + fii* + yi2 + âi + e. FIRST TABLE. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 aj3*yôe 10 100 1000 10000 12 120 1200 12000 20012 111 1110 11100 11012 10102 1002 10020 212 2120 21200 A 1 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 12021 140 ajSySe 20222 2211 22110 21121 11201 12022 20202 2011 20110 1121 11210 12112 21102 11011 10122 1202 12020 20212 2111 21110 A 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 afiySe 11121 11222 12202 22002 20011 101 1010 10100 1012 10120 1212 12120 21212 12111 21122 11211 12122 21202 12011 20122 A 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 ajSySe 12111 12110 21112 11111 11122 11202 12002 20002 11 110 1100 11000 10012 102 1020 10200 2012 20120 1221 12210 A 1 81 82 83 84. 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 ajSyôe 221121 21111 11101 11022 10202 2002 20020 221 2210 22100 21021 10201 2022 20220 2221 22210 22121 21201 12001 20022 A 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 1 116 117 1.18 119 120 | 121 ajSySe 1 211 1 211ft 21100 11021 10222 2202 22020 20221 2201 22010 20121 1201 12010 20112 1111 11110 11112 11102 11002 10002 2 1 SECOND TABLE. A 1 242 1 69 5 2 46 74 70 10 209 6 138 222 3 207 15 47 214 49 75 afiyS* 1 10 11 12 100 1011 102 110 111 112 120 121 122 1000 1001 1002 1010 1011 1012 1020 A 198 149 71 227 230 11 115 216 210 30 143 7 112 36 1139 61 51 223 79 160 aj3y5e 1021 1022 1100 1101 1102 1110 1111 1112 1120 1121 1122 1200 1201 1202 1210 1211 1212 1220 1221 A 4 189 120 208 221 73 16 afiySe 10000 10001 10002 10010 10011 10012 10020 130 10021 166 48 148 14 215 142 229 50 159 35 76 1222 1 92 10022 10100 10101 10102 10110 10111 10112 10120 10121 10122 10200 10201 1 A 85 199 181 232 150 235 105 72 165 119 228 34 13 231 104 84 12 83 118 116 ajSySe 102021 10210 10211 10212 10220 10221 10222 11000 11001 11002 11010 11011 11012 11020 11021 11022 11100 11101 11102 11110 A 64 117 217 41 65 211 25 66 31 56 218 144 202 42 8 99 67 113 59 212 aj8y8e mill 11112 11120 11121 111221 11200 11201 11202 11210 11211 11212 11220 11221 11222 12000 12001 12002 12010 12011 12012 A 1 37 20 1 26 140 179 219 62 54 32 52 174 57 224 154 43 80 176 145 161 184 |203 ajSySe 1 12020 1 12021 12022 12100 12101 12102 12110 12111 12112 12120 12121 12122 12200 12201 12202 12210 12211 12212 12220 12221 12222 1910.] GALOIS FIELD TABLES. 191 OF [28], i8 = i* + i3 + i2 + 1, modulo 2. Ï* = ai1 + /ft6 + yi5 + ôi* + etf + & + tfi + 6. FIRST TABLE. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 J 42 1 43 j 44 45 46 47 J 48 afiy8eÇY)0 10 100 1000 10000 100000 1000000 10000000 11101 111010 1110100 11101000 11001101 10000111 10011 100110 1001100 10011000 101101 1011010 10110100 1110101 11101010 11001001 10001111 11 110 1100 11000 110000 1100000 11000000 10011101 100111 1001110 10011100 100101 1001010 10010100 110101 1101010 11010100 10110101 1110111 11101110 11000001 10011111 100011 1000110 A 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 apySeÇriO 10001100 101 1010 10100 101000 1010000 10100000 1011101 10111010 1101001 11010010 10111001 1101111 11011110 10100001 1011111 10111110 1100001 11000010 10011001 101111 1011110 10111100 1100101 11001010 10001001 1111 11110 111100 1111000 11110000 11111101 11100111 11010011 10111011 1101011 11010110 10110001 1111111 11111110 11100001 11011111 10100011 1011011 10110110 1110001 11100010 11011001 A 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 ;L36 137 138 139 140 j 141 142 143 144 apySeÇy)6 10101111 1000011 10000110 10001 100010 1000100 10001000 1101 11010 110100 1101000 11010000 10111101 1100111 11001110 10000001 11111 111110 1111100 11111000 11101101 11000111 10010011 111011 1110110 11101100 11000101 1Ó010111 110011 1100110 11001100 10000101 10111 101110 1011100 10111000 1101101 11011010 10101001 1001111 10011110 100001 1000010 10000100 10101 101010 1010100 10101000 A 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 ajSySe^rjo 1001101 10011010 101001 1010010 10100100 1010101 10101010 1001001 10010010 111001 1110010 11100100 11010101 10110111 1110011 11100110 11010001 10111111 1100011 11000110 10010001 111111 1111110 11111100 11100101 11010111 10110011 1111011 11110110 11110001 11111111 11100011 11011011 10101011 1001011 10010110 110001 1100010 11000100 10010101 110111 1101110 11011100 10100101 1010111 10101110 1000001 10000010 Ox H-" O o o o o H-« H-" 00 H-• H-J H-' ^r O H-» H-' O 00 o H-' K) o o o o K> HH- 00 O H-' H-' H-< O O* o o H-' ÇO 03 O o H-" KJ 00 o o o H-J K> zo o H-i H-J tO OO O O H-" H-J O K-J •H- H-" O O •""' Oi to o H-> o o H-' ^ o o H-J M N> N> H> O o o H-1 o o o o o H-J ^ o o o o ^1 Ox H-" H-J H-" H-• co co H-' H-> o H-i o n^ H-« O H-" N> ^ 1—I o o tO 00 00 o !—• H-' Ox H-' O H-' O N> K) 00 o o •"' 00 o o o H-J CO 00 H-J H-" tO Oi H-" O Ox o H-' o H-J to H-J o o K> Ox H-J H-' H-' O N> o» Ül H-i ?" 9 Of O) WN ci «. tO H-« l-J H-" H-' tO H-" H (O M K) |_J H-J H-" H-J H-' tO tO H-J H-J © M K) <( N) 0)OQ0l^MOKlC000K)05MM^H^i|XÜlWHMK)^O OOrf^Q0^OC©HH^H-'(^C0OO0trf^H-'C0OO00O00t0'<I0000Cn0iCn^IH- 100 O o 8 00 O H-' H-' H-" Oi H> 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[Jan., OF [73], iB == i -f 5, modulo 7. i\ = aï* + 0î + y. FIRST TABLE. SECOND TABLE. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 «0Y 10 100 15 150 515 134 355 511 164 655 542 404 16 160 615 142 435 326 221 233 353 A 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 «0Y 561 664 632 312 151 525 234 363 661 602 12 120 215 103 45 450 546 444 416 136 305 A 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 114 171 | 228 285 342 «0Y 1 11 110 115 165 665 642 412 166 605 42 420 246 413 106 5 4 6 2 3 1 A 342 228 285 114 57 171 1 43 32 264 280 3 13 2 240 222 35 99 202 56 J 44 a0y 1 2 3 4 5 ' 6 10 11 12 13 14 15 16 100 101 102 103 104 105 106 110 A 267 218 194 133 45 201 33 86 181 168 277 268 283 265 290 75 293 6 219 41 1 281 «0Y 111 112 113 114 115 116 120 121 122 123 124 125 126 130 131 132 133 134 135 136 140 A 78 16 307 64 195 148 4 26 142 82 323 134 296 14 312 186 245 9 46 50 «0Y 141 142 143 144 145 146 150 151 152 153 154 155 156 160 161 162 163 164 165 166 OF [192], i2 s i + 17, modulo 19. ** = ai -f- fi. FIRST TABLE. SECOND TABLE. A 1 2 3 4 5 6 7 8 9 10 11 12 13 a , 1 , 1 , 18 , 16 , 18 , 5 , 7 , 16 , 2 8 4 7 18 0 1 0 17 17 2 6 2 9 5 6 15 3 11 , 5 A 14 15 16 17 18 19 20 40 60 80 100 120 140 a 4 , 6 , 17 5 9 18 0 2 11 7 4 9 1 2 4 8 16 13 7 14 A 160 180 200 220 240 260 280 300 320 1340 360 9 18 17 15 11 3 6 12 5 10 1 t A 360 20 260 40 320 280 120 60 160 340 240 300 100 a,0 1 2 3 4 5 6 7 8 9 10 11 12 13 A 140 220 80 200 180 1 218 183 349 247 95 176 252 a 1 1 1 1 1 1 1 1 > * 14 15 16 17 18 0 1 2 3 > 4 , 5 » 6 , 7 A 46 310 334 288 284 185 193 331 57 2 199 « , 0 1,8 1,9 1 , 10 1 ,11 1 • 12 1 , 13 1 , 14 1 , 15 1 , 16 1 , 17 1 , 18 1910.] GALOIS FIELD TABLES. 195 G-FCa9], i* = i* +1* + i» + i» + t + l, modulo 2. i\ = ad% + /?i* + yi* + ôtf> + et* + #» -f ^ -fc Oi + K. FIRST TABLE. À 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 j 25 26 27 28 29 30 31 32 33 1 34 35 36 j 37 38 39 40 41 42 43 44 45 J 46 1 47 48 apy 8 e ÇriOtc 1 10 100 1000 10000 100000 1000000 10000000 100000000 100011111 100100001 101011101 110100101 1010101 10101010 101010100 110110111 1110001 11100010 111000100 10010111 100101110 101000011 110011001 101101 1011010 10110100 101101000 111001111 1 10000001 100000010 100011011 100101001 101001101 110000101 10101 101010 1010100 10101000 | 101010000 110111111 1100001 11000010 110000100 10111 101110 1011100 10111000 1 101110000 A I 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 1 96 ajSySe^rjÖK 1 111111111 11100001 111000010 10011011 100110110 101110011 111111001 11101101 111011010 10101011 101010110 110110011 1111001 11110010 111100100 11010111 110101110 1000011 10000110 100001100 100000111 100010001 100111101 101100101 111010101 10110101 101101010 111001011 10001001 100010010 100111011 101101001 111001101 10000101 100001010 100001011 100001001 100001101 100000101 100010101 100110101 101110101 111110101 11110101 111101010 11001011 110010110 | 110011 A I 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 ! 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 1 134 I 135 136 137 138 139 140 141 142 143 1 144 a/SySe^rjÔK 1 1100110 11001100 110011000 101111 1011110 10111100 101111000 111101111 11000001 110000010 non 110110 1101100 11011000 110110000 1111111 11111110 111111100 11100111 111001110 10000011 1Ô0000110 100010011 j 100111001 101101101 111000101 10010101 100101010 101001011 110001001 1101 11010 110100 1101000 11010000 110100000 1011111 10111110 101111100 111100111 11010001 110100010 1011011 10110110 101101100 111000111 —1-aj8YÔe£îj0K 1 145 101011011 1 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 10010001 1 191 100100010 1 192 110101001 1001101 10011010 100110100 101110111 111110001 11111101 111111010 11101011 111010110 10110011 101100110 111010011 10111001 101110010 111111011 11101001 111010010 10111011 101110110 111110011 11111001 111110010 11111011. j 111110110 11110011 111100110 11010011 110100110 1010011 10100110 101001100 110000111 10001 100010 1000100 10001000 100010000 100111111 101100001 111011101 10100101 101001010 110001011 1001 10010 1 100100 196 GALOIS FIELD TABLES. [Jan., GFm i9 = i8-H* + i3 + t2 + t + l, modulo 2. ÎK = ai8 + pi1 + yi6 + W + ei4 + #3 + vi2 + 0i + tc. FIEST TABLE.—Continued. A 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 af}y8eÇri0K 1001000 10010000 100100000 101011111 110100001 1011101 10111010 101110100 111110111 11110001 111100010 11011011 110110110 1110011 11100110 111001100 10000111 100001110 100000011 100011001 100101101 101000101 110010101 110101 1101010 11010100 110101000 1001111 10011110 100111100 101100111 111010001 10111101 101111010 111101011 11001001 110010010 111011 1110110 11101100 111011000 10101111 101011110 110100011 1011001 10110010 101100100 111010111 A 1 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 afiy8eÇr)0K 1 10110001 101100010 111011011 10101001 101010010 110111011 1101001 11010010 U0100100 1010111 10101110 101011100 110100111 1010001 10100010 101000100 110010111 110001 1100010 11000100 110001000 1111 11110 111100 1111000 11110000 111100000 11011111 110111110 1100011 11000110 110001100 111 1110 11100 111000 1110000 11100000 111000000 10011111 100111110 101100011 111011001 10101101 101011010 110101011 1001001 10010010 A 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 , 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 aPy8eÇrj6K 1 100100100 101010111 110110001 1111101 11111010 111110100 11110111 111101110 11000011 110000110 10011 100110 1001100 10011000 100110000 101111111 111100001 11011101 110111010 1101011 11010110 110101100 1000111 10001110 100011100 100100111 101010001 1101111Q1 1100101 11001010 110010100 110111 1101110 11011100 110111000 1101111 11011110 110111100 1100111 11001110 110011100 loom 1001110 10011100 100111000 101101111 111000001 10011101 A 337 338 339 340 341 342 343 344 345 346 347 348 349 350 ! 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 a(Zy8eÇr)0K 1 100111010 101101011 111001001 10001101 100011010 100101011 101001001 110001101 101 1010 10100 101000 1010000 10100000 101000000 110011111 100001 1000010 10000100 100001000 100001111 100000001 100011101 100100101 101010101 110110101 1110101 11101010 111010100 10110111 101101110 111000011 10011001 100110010 101111011 111101001 11001101 110011010 101011 I 1010110 10101100 101011000 110101111 1000001 10000010 100000100 100010111 100110001 1910.] GALOIS FIELD TABLES. 197 OF[29], i9 = i* + *4 + & + *2 + * + 1, modulo 2. i* = als 4- pi1 + yi6 -f- ai5 + ei* + ft8 + vi2 + 6Î + K. FIRST TABLE .—Continued. A 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 aPyScCyOic 101111101 111100101 11010101 110101010 1001011 10010110 100101100 101000111 110010001 111101 1111010 11110100 111101000 11001111 110011110 100011 1000110 10001100 100011000 100101111 101000001 110011101 100101 1001010 10010100 100101000 101001111 110000001 11101 111010 1110100 11101000 A 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 apy8e£r)0K 111010000 10111111 101111110 111100011 11011001 110110010 1111011 11110110 111101100 11000111 110001110 11 110 1100 11000 110000 1100000 11000000 110000000 11111 111110 1111100 11111000 111110000 11111111 111111110 11100011 111000110 10010011 100100110 101010011 110111001 A 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 afiy8eÇri$ K 1101101 11011010 110110100 1110111 11101110 111011100 10100111 101001110 110000011 11001 110010 1100100 11001000 110010000 111111 1111110 11111100 111111000 11101111 111011110 10100011 101000110 110010011 111001 1110010 11100100 111001000 10001111 100011110 100100011 1 101011001 110101101 A 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 1 511 apySeÇriBic 1 1000101 10001010 100010100 100110111 101110001 111111101 11100101 111001010 10001011 100010110 100110011 101111001 111101101 11000101 110001010 1011 10110 101100 1011000 10110000 101100000 111011111 10100001 101000010 110011011 101001 1010010 10100100 101001000 110001111 1 198 GALOIS FIELD TABLES. [Jan., GF[29], i9 = i8+i*-\-i3 + i2+i+ly modulo 2. ik = ai8 -f f3tf -f yi* -f- di5 -f ei* + Ci3 + W2 + Si + «. SECOND TABLE. m 511 1 428 2 345 429 273 3 190 ! 346 496 430 127 274 262 4 179 191 299 347 35 497 44 431 458 128 107 275 413 263 436 5 i ajS-y SC^TJÔK 1 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 10001 10010 10011 10100 10101 10110 10111 11000 11001 11010 non 11100 11101 11110 11111 100000 A I 353 180 400 192 407 300 330 348 506 36 1 375 498 24 45 100 432 258 459 96 129 216 108 320 276 472 414 230 264 394 437 463 6 apy8eÇr)$K 1 100001 100010 100011 100100 100101 100110 100111 101000 1Ô1001 101010 101011 101100 101101 101110 101111 110000 110001 110010 j 110011 110100 110101 110110 110111 111000 111001 111010 111011 111100 111101 111110 111111 1000000 \ 1 380 354 66 181 481 401 311 193 287 ! 408 389 301 147 331 220 349 254 507 175 37 13 376 250 499 237 25 139 46 198 101 133 433 aj3y$e£i}0ie 1 1000001 1000010 1000011 1000100 1000101 1000110 1000111 1001000 1001001 1001010 1001011 1001100 1001101 1001110 1001111 1010000 1010001 1010010 1010011 1010100 1010101 1010110 1010111 1011000 1011001 1011010 1011011 1011100 1011101 1011110 1011111 1100000 À 1 41 259 270 460 317 97 327 130 247 217 308 109 449 321 324 277 17 473 206 i 415 363 231 452 265 61 395 423 438 292 464 112 7 apy8eÇr)ÔK 1100001 1100010 1100011 1100100 1100101 1100110 1100111 1101000 1101001 1101010 1101011 1101100 1101101 1101110 1101111 1110000 1110001 1110010 1 1110011 1110100 1110101 1110110 1110111 1111000 1111001 1111010 1111011 1111100 1111101 1111110 1111111 10000000 1910.] GALOIS FIELD TABLES. 199 G-F[29], i*sstf» + ^-|-** + *, + î + l, modulo 2. i\ = ai8 -f £i7 + yi6 + ôtf + si* + ci8 + rçi2 + 0t' -f K. SECOND TABLE.—Continued. 29 ' 381 117 355 82 67 209 182 77 482 489 402 340 312 476 194 143 288 445 409 123 390 20 302 369 148 52 332 336 221 280 350 503 255 469 508 187 176 455 38 244 14 58 377 284 251 234 500 aj8ySe£ï?0K 10000001 10000010 10000011 10000100 10000101 10000110 10000111 10001000 10001001 10001010 10001011 10001100 10001101 10001110 10001111 10010000 10010001 10010010 10010011 10010100 10010101 10010110 10010111 10011000 10011001 10011010 10011011 10011100 10011101 10011110 10011111 10100000 10100001 10100010 10100011 10100100 10100101 10100110 10100111 10101000 10101001 10101010 10101011 10101100 10101101 10101110 10101111 10110000 A 241 238 156 26 74 140 366 47 159 199 164 102 225 134 418 434 105 42 297 260 494 271 426 461 228 318 94 98 373 328 398 131 137 248 173 218 387 309 64 110 421 450 204 322 306 325 268 278 a(iy8eÇri0K 10110001 10110010 ! 10110011 10110100 10110101 10110110 10110111 10111000 10111001 10111010 10111011 10111100 10111101 10111110 10111111 11000000 11000001 11000010 11000011 11000100 11000101 11000110 11000111 11001000 11001001 11001010 11001011 11001100 11001101 11001110 11001111 11010000 11010001 11010010 11010011 11010100 11010101 11010110 11010111 11011000 11011001 11011010 11011011 11011100 11011101 11011110 11011111 11100000 A 50 18 443 474 487 207 115 416 162 364 154 232 56 453 467 266 202 62 171 396 92 424 295 439 167 293 169 465 152 113 441 8 358 30 211 382 87 118 69 356 85 83 84 68 86 210 357 183 a&y8èÇv)9K 11100001 11100010 11100011 1 11100100 11100101 11100110 11100111 11101000 11101001 11101010 11101011 11101100 11101101 11101110 ! molm 11110000 11110001 11110010 11110011 11110100 11110101 11110110 11110111 11111000 11111001 11111010 11111011 11111100 11111101 11111110 11111111 100000000 100000001 100000010 100000011 100000100 100000101 100000110 100000111 100001000 100001001 100001010 100001011 100001100 100001101 100001110 100001111 100010000 A 70 78 119 483 88 490 383 403 212 341 31 313 359 477 9 195 10 144 478 289 360 446 314 410 32 124 342 391 213 21 404 303 384 370 491 149 89 53 1 484 333 120 337 79 222 71 281 1 184 351 100010001 100010010 100010011 100010100 100010101 100010110 100010111 100011000 100011001 100011010 100011011 100011100 100011101 100011110 100011111 100100000 100100001 100100010 100100011 100100100 100100101 100100110 100100111 100101000 100101001 100101010 100101011 100101100 100101101 100101110 100101111 100110000 100110001 100110010 100110011 100110100 100110101 100110110 100110111 100111000 100111001 100111010 100111011 100111100 100111101 100111110 100111111 101000000 200 GALOIS FIELD TABLES. [Jan., GF[29], i9 m t* + Î* + i8 + & + i 4-1, modulo 2. fcA = Cti8 -f /^ -f 7i6 _j_ fifi _|_ ^4 _|_ ^3 _|_ V{2 _|_ ft -f K. SECOND TABLE.—Continued. A 405 504 22 256 214 1 470 392 509 i 343 188 125 177 33 456 411 39 315 245 447 15 361 59 290 378 479 285 145 252 11 235 196 501 185 242 282 239 72 157 223 27 80 75 338 141 121 367 334 48 apy8e$ri9K 101000001 101000010 101000011 101000100 101000101 101000110 101000111 101001000 101001001 101001010 101001011 101001100 101001101 101001110 101001111 101010000 101010001 101010010 101010011 101010100 101010101 101010110 101010111 101011000 101011001 101011010 101011011 101011100 101011101 101011110 101011111 101100000 101100001 101100010 101100011 101100100 101100101 101100110 101100111 101101000 101101001 101101010 j 101101011 101101100 101101101 101101110 101101111 1 101110000 A 1 485 160 1 54 200 90 165 150 103 | 492 226 371 135 385 419 304 435 412 106 457 43 34 298 178 261 126 495 189 272 344 427 510 462 393 229 471 319 | 215 95 1 257 99 23 374 . 505 329 406 399 352 1 132 aj8y8e£i70K I 101110001 101110010 101110011 101110100 101110101 101110110 101110111 101111000 101111001 101111010 101111011 101111100 101111101 101111110 101111111 110000000 110000001 110000010 110000011 110000100 110000101 110000110 110000111 110001000 110001001 110001010 110001011 110001100 110001101 110001110 110001111 110010000 110010001 110010010 110010011 110010100 110010101 110010110 110010111 110011000 110011001 110011010 110011011 110011100 110011101 110011110 110011111 1 110100000 A 1 197 138 236 249 12 174 253 219 146 388 286 310 480 65 379 111 291 422 60 451 362 205 16 323 448 307 246 326 316 269 40 279 335 51 368 19 122 444 142 475 339 488 76 208 81 116 28 417 aj8y5e£rj0/e lioiooooi ! 110100010 110100011 110100100 110100101 110100110 110100111 110101000 110101001 110101010 110101011 110101100 110101101 110101110 110101111 110110000 110110001 110110010 110110011 110110100 110110101 110110110 110110111 110111000 110111001 110111010 110111011 110111100 110111101 110111110 110111111 111000000 111000001 111000010 111000011 111000100 111000101 111000110 111000111 111001000 111001001 111001010 111001011 111001100 111001101 111001110 111001111 1 111010000 A 224 163 158 365 73 155 240 233 283 57 243 454 186 468 502 267 305 203 420 63 386 172 136 397 372 93 227 425 493 296 104 440 151 168 1 166 294 91 170 201 466 55 153 161 114 486 442 49 L a&y&eCr)0K 1 111010001 111010010 111010011 111010100 111010101 111010110 111010111 111011000 111011001 111011010 111011011 111011100 111011101 111011110 111011111 111100000 111100001 111100010 111100011 111100100 111100101 111100110 111100111 111101000 111101001 111101010 111101011 111101100 111101101 111101110 111101111 111110000 111110001 111110010 111110011 111110100 111110101 111110110 111110111 111111000 | 111111001 111111010 111111011 111111100 111111101 111111110 111111111 1910.] GALOIS FIELD TABLES. 201 OF [232], i2 = i + 16, modulo 23. i* = ai + /?. FIRST TABLE. SECOND TABLE. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 a , 0 1 1, 0 1,16 17,16 10,19 6,22 5, 4 9,11 20, 6 3,21 1, 2 3,16 19, 2 21, 5 3,14 17, 2 A 16 1 17 18 19 20 21 22 23 24 1 48 72 96 120 144 J168 a , 0| 19,19] 15, 5 1 20,10 7,21 5,20 2,11 13, 9 22, 1 7 3 21 9 17 4 1 5 A 192 216 240 264 288 312 336 360 384 |408 432 456 480 504 |528 a , 0 1 12 15 13 22 16 20 2 14 61 19 18 11 8 10 1 A 528 336 48 144 168 384 24 480 96 504 456 192 240 360 |216 a , (S I 1 2 3 4 5 6 7 8 9 10 11 12 13 A 2P* 120 432 408 312 72 264 1 136 10 1523 380 411 14 1 310 15 J 489 *,0 | 16 1 17 18 19 20 21 22 1, 0 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 |1, 7 A | 329 1 469 366 132 234 491 439 423 2 213 28 149 494 224 |287 a , (3 1, 8 1, 9 1,10 1,11 1,12 1,13 1,14 1,15 1 1,16 1 1,17 1,18 1 1,19 1,20 1,21 iiil GF [5*], t* = t» + i 4- 2, modulo 5. it = ai3 + pi* + yi + ô% FIRST TABLE. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 apyS 10 100 1000 1012 1132 2332 344 3440 2431 1334 4302 2013 2104 3014 3121 4241 1403 42 420 4200 1043 1442 432 4320 2243 4404 3033 A 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 1 54 ajSyô 3311 1141 2422 1244 3402 2001 2034 2314 114 1140 2412 1144 2402 1044 1402 32 320 3200 31 310 3100 4031 4303 2023 2204 4014 4133 A 55 56 57 58 59 60 61 62 63 64 m m 67 68 69 70 71 72 73 74 75 76 77 78 79 80 1 81 a/3yô 323 3230 331 3310 1131 2322 244 2440 1424 202 2020 2224 4214 1133 2342 444 4440 3443 2411 1134 2302 44 440 4400 3043 3411 2141 A 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 afiyS 3434 2321 234 1 2340 424 4240 1443 442 4420 3243 411 4110 143 1430 312 3120 4231 1303 4042 4413 3123 4211 1103 2042 2444 1414 A 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 102 j135 afiyS 1020 1212 3132 4301 2003 2004 2014 2114 3114 4121 203 2030 2324 214 2140 3424 2221 4234 1333 4342 2413 1104 2002 2044 2414 1114 2102 A 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 312 468 624 apy& 3044 3421 2241 4434 3333 1311 4122 213 2130 3324 1221 3222 201 2010 2124 3214 121 1210 3112 4101 3 4 2 1 202 GALOIS FIELD TABLES. [Jail., OF [54], i* = i3 + i + 2, modulo 5. i* = ai3 + /?# + yi + â. SECOND TABLE. A 624 468 156 312 1 388 514 330 511 2 220 108 304 275 389 382 525 401 36 515 152 217 335 564 331 398 278 ajSyS 1 2 3 4 10 11 12 13 14 100 101 102 103 104 110 111 112 113 114 120 121 122 123 124 130 131 132 A 475 299 512 523 240 94 404 3 287 270 189 269 221 495 4 412 207 109 261 288 482 361 305 547 271 604 168 ajSyÔ 133 134 140 141 142 143 144 1000 1001 1002 1003 1004 1010 1011 1012 1013 1014 1020 1021 1022 1023 1024 1030 1031 1032 1033 1034 A 276 365 190 21 41 390 338 208 104 130 383 608 222 281 134 526 451 496 294 181 402 59 5 68 74 37 29 a|3yÔ 1040 1041 1042 1043 1044 1100 1101 1102 1103 1104 1110 1111 1112 1113 1114 1120 1121 1122 1123 1124 1130 1131 1132 1133 1134 1140 1141 A 413 613 39 516 196 362 323 424 153 186 110 440 579 218 146 262 585 622 336 194 289 229 285 565 570 483 165 afiyS 1142 1143 1144 1200 1201 1202 1203 1204 1210 1211 1212 1213 1214 1220 1221 1222 1223 1224 1230 1231 1232 1233 1234 1240 1241 1242 1243 A 31 332 291 169 99 500 399 141 306 550 328 279 438 548 237 410 476 540 272 127 10 300 379 605 592 415 513 ajSyO 1244 1300 1301 1302 1303 1304 1310 1311 1312 1313 1314 1320 1321 1322 1323 1324 1330 1331 1332 1333 1334 1340 1341 1342 1343 1344 1400 A 231 42 17 467 524 216 277 239 107 241 225 366 619 63 95 559 191 454 430 405 162 22 88 615 a/3yô 1401 1402 1403 1404 1410 1411 1412 1413 1414 1420 1421 1422 1423 1424 1430 1431 1432 1433 1434 1440 1441 1442 1443 1444 OF [36], i6 = i -f 1, modulo 3. ik =. aib + Pi* + 7i* + ê& + ei -f C FIRST TABLE. A 1 2 3 4 5 6 7 8 9 10 11 12 13 a/3y3e£ | 10 | 100 1000 10000 100000 11 110 1100 11000 110000 100011 121 | 1210 A 14 15 16 17 18 19 20 21 22 23 24 25 26 apy8e£ 12100 ' 121000 210011 100102 1001 10010 100100 1011 10110 101100 11011 110110 101111 A 27 28 29 30 31 32 33 34 35 36 37 38 1 39 afiySeÇ 11121 111210 112111 121121 211221 112202 122001 220021 200202 2012 20120 201200 12022 A 40 41 42 43 44 45 1 46 47 48 49 50 51 I 52 aPySeÇ 120220 202211 22102 221020 210222 102212 22101 221010 210122 101212 12101 121010 210111 A 53 54 55 56 57 58 59 60 61 62 j 63 64 65 a&y8eÇ 1 101102 11001 110010 100111 1121 11210 112100 121011 210121 101202 12001 120010 200111 1910.] GALOIS FIELD TABLES. 203 OF [36], i* = i + 1, modulo 3. ^ -=<n5 + /?** + 7*3 + W + e* + C FIRST TABLE.—Continued. A 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 J 113 apySeÇ 1 1102 11020 110200 102011 20121 201210 12122 121220 212211 122102 221001 210002 100012 101 1010 10100 101000 10011 100110 1111 11110 111100 ! 111011 110121 101221 12221 122210 222111 221102 211012 110112 101101 11021 110210 102111 21121 211210 112122 121201 212021 120202 202001 20002 200020 222 2220 22200 222000 A 114 115 116 117 1X8 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 1161 afiydeÇ 220022 200212 2112 21120 211200 112022 120201 202021 20202 202020 20222 1 202220 22222 222220 222222 , 222212 222112 221112 211112 111112 111101 111021 110221 102221 22221 222210 222122 221212 212112 121112 211101 111002 110001 100021 221 2210 22100 221000 210022 100212 2101 21010 210100 101022 10201 102010 20111 2Ö1110 A 1 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 |209 aj8->/ôe£ 11122 111220 112211 122121 221221 212202 122012 220101 201002 10012 100120 1211 12110 121100 211011 110102 101001 10021 100210 2111 21110 211100 111022 110201 102021 20221 202210 22122 221220 212222 122212 222101 221002 210012 100112 1101 11010 110100 101011 10121 101210 12111 121110 211111 111102 111001 110021 J 100221 A 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 (257 aj8ySe£ 1 2221 22210 222100 221022 210212 102112 21101 211010 110122 101201 12021 120210 202111 21102 211020 110222 102201 22021 220210 202122 21212 212120 121222 212201 122002 220001 200002 12 120 1200 12000 120000 200011 102 1020 10200 102000 20011 200110 1122 11220 112200 122011 220121 201202 12012 120120 1 201211 A 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 |305 aflySeÇ 1 12102 121020 210211 102102 21001 210010 100122 1201 12010 120100 201011 10102 101020 10211 102110 21111 211110 111122 111201 112021 120221 202221 22202 222020 220222 202212 22112 221120 211222 112212 122101 221021 210202 102012 20101 201010 10122 101220 12211 122110 221111 1 211102 111012 110101 101021 10221 102210 22111 1 H—' I --4 H-J 1 00 00 I 1—" H-" 1 to to I l—" H-' i- O 1 H-i to co h-" 1—" o o f-1 O 1 tO H-J o ^ 00 rfs- to to H-J H-• H-J I—" i— O I Ci H-» -q QO ZO O I o o f o o to to 1—i H-» i— O •—' œ> en zo zo o o to to o o 1—» 1—1 i— O Ül H-" 00 1—» to o to Ci ^4 H-J o to tO Ül oo N> H-• O to CO ** Ci O O ro o to o oo Ül o to o o to N> Ci Ül t—' tO O Ü» O Ül o K> O 1—1 00 to co o o to o •"•' ÜI 00 >^ o to o o l-i N> 00 o 1—« to o o H-" o o ^ to H-• o o ^4 1—1 00 o o tO o o tO t*> o o to o o o to rfj* zo H-' to to Ül K) ^ o tO to oo zo N> o to to tO Ci ^ O o H-' to to Üi oo Ci o H-' K) to to on ^4 i—' K) l—" 00 o o to H-' to K> O tO O to H-> rf^ --I CO o o H-" to 1—• CO o o I—" K> to H-' Ox H-> oo 1—• to c> oo Ox © to o OX H-' CO to o to o H-' -«J to o o 1—I to o to CO Ox © to ro o Ox ^4 ^ 1—• to rf^ 00 o tO to tO Ox Ox ro O H-• to ^ co oa o o i—• H-" to h£t co o to to 00 Ox H-' H-' tO ^4 o to H-' Ci co -v| tO o H-J H-J Ox c& o o h-1 H-' Ül co oo o to H-J I—1 rf^ —1 Ox H-' © £» o o to o to Ci Ci to o I—1 o GO t^ o o 1—• H-' o K) o to o K) H-1 o Ci os I—1 o to o Ox o tO to Ci to Ci to g to I—" ^1 o o o to Ci to o 1—1 to o to ^ CO 1—1 o «—• ,_, Ox © to I—1 Ci oo ro o o H-1 00 CO co g I—" o 1—1 to co o K) o 1—• GO 1—I o o K) ^ o to o K) ^ o to o g to o o o o o Ci CO Oi o tO o o oo Ox o (—» o K> to tO co o H-J to Ci Ci CO ro to to Ci o Ci o o o to to ^ o o» o N) to rf^ o o H-1 o tO H-J N> O o Ci 4^ 00 to to ,__, ^ ^4 o o o to H-' Ci rf^ ^4 O 1—' H-• tO H-J to ^ rfj* o tO O Ci co O H-< O to Ox © K) ro o OO N) OO O o o to o Ox Ox to o to o Ci CO tf*. H-J O to Ox Ox O i—' to oo Ox Ci K> 1—' to ^4 00 o o o ^—' to Ci ^ OO O h-J to K) 1—' 1—• o t—> ^ 00 00 o h-" 1-1 Ox Ox 00 K> 1-1 H-' H-' O o o 1—• hO Ci O 1—« 1—< H-' 1—" 00 o o 1—1 o to to o 1—' o Ox 00 hO I—1 o rf^ ^4 oo o o o o rf-00 co o 1—" H-' o ^1 H-" 1—1 o o to to Ci CO o I—1 o to to t—' o to o to -^ K) -^ O o o o to Ox 00 o t—1 o to 1—' 00 o o H-" ^ 00 Ci o o 1—1 ^ o Ci to o H-" Ci o o o o o o h—' co *4 o 1—' o i—• oo 1—' o o o GO 1-1 o o o Ox t—' ^ to o o Ox o o o o o K) 00 o 1—• 1—" o o Ci 00 00 H-» to to Ci H-' •"-' o o to to ,_, Ci to to to Ci 00 ^1 to to to to ,_, Ox ^4 o H-< o hO to to 1—1 to H-" 1—' -vl co o o K) H-• K) ^4 K) H-' co t—' K) to K) t—' 00 O to o I—" o to H-" to 00 00 1—' K> O ^4 to O o to o Ül ^4 Ül hO o Ül ^ Ci to to to o to ^4 O o H-' o to o Ül H-" to H-" H-" to ,_, ^4 H-< O o to Ül o to H-" to ^ Ci Ül ro to to *» oo Ül O H-" O H-" to ^ ^4 rf^ H-" H-" 00 oo o o H-" tf^ co o 1-1 to co Ci to K) H-' K) O O o H- o H-' H-' *4 H-' O CO o o t—' o 00 Ci O rf^ GO *"" to to o rf^ 00 ^4 O H-« O H-" O K) ^^ 00 H-' O to 00 to to o o o to «<! O o o to Ül 00 o J° hO o to rf>--v| H-' o I—1 s to «^J co H-" o H-" tf^ ^4 to o o o H-' Ci f^ tf^ o H-' Ül oo -^1 to ro o ^ H-" ^4 00 o H-" o o H-' to H-" o o rf^ o o o o rf^ ^4 Ci o o Ci co o K) hO o o 00 to o H-' o o o K> 00 ^1 to Ül rf>* Ül bO to to ^4 -vl o K) to •^4 to KJ H-" K> to h^ h0 CO o o to to to Ci H-' H-" rf^ 00 o H-J hO to H-* CO 00 o to H-' Ül co ^ hÜ K) H-' hO o CO o o to to (—• H-* H-" o Ci 00 co (—• to hO o Ci ^4 O hO o oo H-' ^4 to hO O Ci H-1 to o o KJ 8 -a to 00 H-1 00 H-i Ci H-< K> H-" to Ül co H-' O H-J to 00 •^ O to H-" to ,_, Ül 00 o O to to > «4 09 1 5T\ > P I T3D 1 ^ 1 O? 1 ir\ 1 > I ^ 1 XD 1 "^ 1 °» 1 <f\ 1 >- 1 ^ 1 "Cö 1 -^ 1 <r\ 1 > 1 ^ 1 -co 1 Oo 1 A 1 ft O o E? 1 00 OO I H-" H-" I ^4 Ci 1 ,--' tO H-« t— tO tO H-" o to I 00 00 to to CO GO H-• H-' O H-" o o to o o to H-J tO 1 00 00 tf^ rf^ H-" O H-" tO H-i H-- tO H-" H-" tO H-" o to 1 00 00 Ül Ül oo to •—' to o to to o to H-" O H-> O 1 00 Ci ri^ to 00 H-' Ül hO o o t—* to to oo hO -4 to o o o 00 oo CO N> o H-' ro H-> to oo Ül H-' t—> o K) to o oo Ci 00 h0 o o o o H-' 00 H-" >+^ to to o o H-> o 00 to Ci ro H-" H-J o o oo oo oo N> to o to to oo Ül o H-• o K> to oo Ci to H-' hu o o o to oo H-" 00 N> to o o H-J 00 to Ül K) H-J H-» o oo oo -*4 N> to to o H-' o 00 M^ co N> s H-" o o 00 Ci H-J N> H-" ro o o H-" 00 to H-' o to to o to 00 to ^ r>ü H-J H-" 00 00 Ci ro to ro o H-" oo t^ 00 K) O o H^ o 00 Ci o t—' to H-J ro o to 00 H-J H-i K> O ro ro H-' 00 K) OO H-« O o o to o oo 00 Ox H-' o ro to K) to oo rf^ ^4 ro o o H-« 00 Ül co ro H-J K) H-J to H-J 00 O H-1 ro H-J O ro H-" 00 to to H-> o o o to oo OO »J> K) H-> O ro to o oo ^ Ci H-J O o K> O to 00 Ül 00 H-J to H-' N) t—* H-" 00 o CO H-J H-" ro H-J O H-J 00 K) H-J to o H-J o o H-J 00 00 00 K) H-J O K) to OO ^ Ül ro H-J O o to H-J 00 Ül ^4 H-J K> H-" K) O 00 o 00 H-" H-J H-J K> H-i to 00 to o ,_, ro o H-" O to 00 00 to ro o K> H-J O o 00 M^ £t H-> ro H-J O O H-" 00 Ül Ci H-J 1—» K> h-J to 00 o -~4 ro H-' H-' H-• to to 00 co to (—• ro o H-" H-J 00 00 H-' N> O ro H-« O oo f^ OO H-• H-J K) H-" O to 00 Ül Üt to o H-J hO to oo o Ci to ro H-J H-J H-J O 00 H-J GO H-* to ro o o oo oo o to g t—' 00 Hi- to ro H-" H-• ro H-' H-J 00 Ül >J> ro ro o H-' H-> O > p TjD "^ o> A «N ^ "CO -^ «N > » "CO 1 -4 1 Oo 1 A 1 «N 1 > » •co 1 *^ 1 o> I A 1 o\ 1 £ 1 "CO I *^ 1 o? I WN j fc ^ + to o S g W o dulo ST T fe s° F *l ^ k II §~% 1 + B- Î5 y> + ^ «a + o, + «». + J*^ H iOIS ^ HH W h1 ö CO ft QC f 1 *H SO c 1910.] GALIOS FIELD TABLES. 205 OF [36], t6 = • + 1, modulo 3. ik = at» + /3i* + yï3 + ^2 + et + f. SECOND TABLE.—Continued. A 291 657 186 632 402 618 261 272 100 215 435 703 621 352 226 312 304 719 45 525 137 335 10 146 599 55 478 398 678 208 328 199 301 177 25 646 96 aj3yôe£ 102012 102020 102021 102022 102100 102101 102102 102110 102111 102112 102120 102121 102122 102200 102201 102202 102210 102211 102212 102220 102221 102222 110000 110001 110002 110010 110011 110012 110020 110021 110022 110100 110101 110102 110110 110111 110112 A 592 89 218 68 185 533 1 99 702 617 718 136 225 477 207 145 645 88 300 701 135 184 87 134 206 491 492 133 503 493 275 576 276 557 28 504 308 163 aj8ySe£ 110120 110121 110122 110200 110201 110202 110210 110211 110212 110220 110221 110222 111000 111001 111002 111010 111011 111012 111020 111021 111022 111100 111101 111102 111110 111111 111112 111120 111121 111122 111200 111201 111202 111210 111211 111212 111220 A 494 457 515 | 558 440 407 577 653 411 277 119 59 309 343 554 29 530 357 505 103 251 458 32 649 164 287 670 495 662 241 441 362 64 516 709 627 |559 aj8yÔe£ 1 A 111221 380 111222 112000 112001 112002 112010 112011 112012 112020 112021 112022 112100 112101 112102 112110 112111 112112 112120 112121 112122 112200 112201 112202 112210 112211 112212 112220 112221 112222 120000 120001 120002 120010 120011 120012 120020 120021 267 654 320 698 408 675 256 578 624 520 120 106 221 412 425 40 278 416 15 344 725 51 60 469 259 310 683 175 531 597 204 555 143 341 30 143S aj8 y8e£ 1 120022 120100 120101 120102 120110 120111 120112 120120 120121 120122 120200 120201 120202 120210 120211 120212 120220 120221 120222 121000 121001 121002 121010 121011 121012 121020 121021 121022 121100 121101 121102 121110 121111 121112 121120 121121 121122 A 318 104 360 595 358 723 73 506 232 691 33 234 588 252 168 581 459 540 482 288 75 297 650 395 466 165 706 547 663 508 92 671 192 638 496 569 apyèeÇ 1 121200 121201 121202 121210 121211 121212 121220 121221 121222 122000 122001 122002 122010 122011 122012 122020 122021 122022 122100 122101 122102 122110 122111 122112 122120 122121 122122 122200 122201 122202 122210 122211 122212 122220 122221 122222 206 GALOIS FIELD TABLES. [Jan., GF[292J, i* = i +26, modulo 29, i* = ai + p. FIRST TABLE. SECOND TABLE. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 a , 0 1, 0 1,26 27,26 24, 6 1,15 16,26 13,10 23,19 13,18 2,19 21,23 15,24 10,13 23,28 22,18 11,21 3,25 28,20 19, 3 A 20 21 22 23 24 25 26 27 28 29 30 60 90 120 150 180 210 240 270 a , 0 22, 1 23,21 15,18 4,13 17,17 5, 7 12,14 26,22 19, 9 28, 1 3 9 27 23 11 4 12 7 21 A 300 330 360 390 420 450 480 510 540 570 600 630 660 690 720 750 780 810 840 a , jS 1 5 15 16 19 28 26 20 2 6 18 25 17 22 8 24 14 13 10 1 A 840 510 30 180 300 540 240 690 60 810 150 210 780 750 330 360 630 570 390 a , 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 A 480 270 660 120 720 600 450 90 420 1 234 478 67 200 734 656 532 581 438 a , 0 20 21 22 23 24 25 26 27 28 1, 0 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7 1, 8 1, 9 A 43 741 417 565 195 5 753 69 827 522 469 728 124 706 340 683 2 486 449 « , 0 1,10 1,11 1,12 1,13 1,14 1,15 1,16 1,17 1,18 1,19 1,20 1,21 1,22 1,23 1,24 1,25 1,26 1,27 1,28 GF[W}, i2 = i-{ 19, modulo 31, fr = ai + p. FIRST TABLE. SECOND TABLE. A. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a , /3 1 1, 0 1,19 20,19 8, 8 16,28 13,25 7,30 6, 9 15,21 5, 6 11, 2 13,23 5,30 4, 2 6,14 20,21 10, 8 18, 4 22, 1 23,15 A 2L 22 23 24 25 2d 27 28 29 3(3 3] 32 64 m 128 160 192 224 25G 288 a , /3 7, 3 10, 9 19, 4 23,20 12, 3 15,11 26, 6 1,29 30,19 18,12 30, 1 12 20 23 28 26 2 24 9 15 A 320 352 384 416 448 480 512 544 576 608 640 672 704 736 768 800 832 864 896 928 960 « , 0 1 25 21 4 17 18 30 19 11 8 3 5 29 7 22 16 6 10 27 14 13 1 A a , j3 960 192 608 384 640 800 704 576 256 832 544 32 928 896 288 768 416 448 512 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 A 352 736 96 224 320 160 864 128 672 480 1 388 884 427 150 827 333 145 953 698 a 1 1 1 1 1 1 1 1 1 1 , 18 1 21 22 ! 23 24 25 26 27 28 29 30 , o , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 A 471 542 509 888 530 912 590 168 899 2 !681 38 263 175 243 j 197 330 277 44 28 [511 *, i8 1,10 1 1,11 1,12 1,13 1,14 1,15 1,16 1,17 1,18 1 ,19 1,20 1,21 1,22 1,23 1,24 1,25 1,26 1,27 1,28 1,29 1,30