Best Known (72, 72+∞, s)-Nets in Base 9
(72, 72+∞, 165)-Net over F9 — Constructive and digital
Digital (72, m, 165)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (72, 164)-sequence over F9, using
- t-expansion [i] based on digital (64, 164)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- T4 from the second tower of function fields by GarcÃa and Stichtenoth over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- t-expansion [i] based on digital (64, 164)-sequence over F9, using
(72, 72+∞, 192)-Net over F9 — Digital
Digital (72, m, 192)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (72, 191)-sequence over F9, using
- t-expansion [i] based on digital (61, 191)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 61 and N(F) ≥ 192, using
- t-expansion [i] based on digital (61, 191)-sequence over F9, using
(72, 72+∞, 602)-Net in Base 9 — Upper bound on s
There is no (72, m, 603)-net in base 9 for arbitrarily large m, because
- m-reduction [i] would yield (72, 1805, 603)-net in base 9, but
- extracting embedded OOA [i] would yield OOA(91805, 603, S9, 3, 1733), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 759 426919 030897 795475 117459 657554 255458 892426 991657 456851 975984 659439 808321 434116 314155 501374 939224 265132 033072 854497 858548 234900 066854 765546 643620 885747 643440 696803 412386 740858 546657 550644 145578 633536 019507 437630 551582 576765 172636 864398 126485 404768 827853 973732 281028 881008 338479 828825 052497 135721 299846 159453 920740 713401 411954 346179 789416 789270 536829 871059 930091 403979 779132 747829 055086 929244 233104 387126 247970 069623 684703 616518 206402 586568 613740 852855 912258 288121 910067 532636 159612 396303 930509 826590 193224 388508 891618 927504 816625 488689 642982 700532 041609 688222 274598 883741 930984 201715 739575 677189 721689 918476 715722 195728 013241 302542 544443 566382 525638 842105 695888 862736 490308 808946 258344 215578 913271 397521 913990 351133 939692 837909 093406 782393 246676 744255 743821 945446 560379 570939 722012 875212 881389 646818 513440 622119 582229 640257 985225 338839 385530 019726 451661 126381 039260 384142 211908 734460 710584 972840 163603 774673 521496 634252 775021 786995 311847 825480 944816 761485 263735 638726 341423 185727 261845 989172 792784 444584 278539 518128 513072 514919 751657 052929 097892 186404 844548 030863 987381 192844 068084 152021 208634 275009 934360 609774 000304 176189 269915 326047 740110 430512 652301 078998 496736 022532 881632 654132 836342 570745 897238 636666 373789 769500 810441 767319 408253 524788 707583 870563 812031 126231 003122 752192 263050 516554 599925 743531 885461 555967 076503 355384 654592 239351 248385 589041 905576 886124 557745 808159 778843 264574 897412 732841 252057 281330 184263 002470 584075 084644 847066 595947 056524 234181 556097 942404 939498 730329 590231 534018 595898 688228 177637 414505 924689 183769 015025 591599 693809 202246 103554 785435 216988 233827 313851 279658 842266 125546 612402 181988 338599 960472 588657 225808 399953 996642 772496 983644 016408 955854 298457 311990 768569 162224 042854 259648 656629 513553 / 289 > 91805 [i]
- extracting embedded OOA [i] would yield OOA(91805, 603, S9, 3, 1733), but