Best Known (239, 239+∞, s)-Nets in Base 3
(239, 239+∞, 112)-Net over F3 — Constructive and digital
Digital (239, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (239, 111)-sequence over F3, using
- base reduction for sequences [i] based on digital (64, 111)-sequence over F9, using
- s-reduction 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
- s-reduction based on digital (64, 164)-sequence over F9, using
- base reduction for sequences [i] based on digital (64, 111)-sequence over F9, using
(239, 239+∞, 163)-Net over F3 — Digital
Digital (239, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (239, 162)-sequence over F3, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 151 and N(F) ≥ 163, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
(239, 239+∞, 493)-Net in Base 3 — Upper bound on s
There is no (239, m, 494)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (239, 2956, 494)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32956, 494, S3, 6, 2717), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 565 517428 561404 409210 688748 729438 720525 385641 151589 531013 274001 274038 603427 757666 130915 529403 023184 756209 956576 424922 883352 360535 346462 846842 395248 583540 140631 531456 188554 399324 015409 446443 345916 644760 559254 263113 029331 314569 969595 286701 135083 464973 227036 388474 889531 500412 403554 570882 846332 906158 848596 598163 757969 862031 070829 510890 114199 202899 751591 707366 170254 855632 655160 198499 711158 259647 683415 088575 813576 015171 017435 000361 522209 318909 472933 950066 220127 830329 983188 427147 677364 487030 773924 488700 955239 915146 000874 868018 167902 662535 083390 237466 929617 873607 967822 734547 838352 763583 830071 291893 277894 171183 743512 533063 563673 170353 208551 008440 357025 389339 339355 161742 857364 079481 087297 747588 149694 443573 574544 604818 603568 355601 763219 832571 677631 541545 233913 388589 130841 512627 280026 643342 386237 676137 918200 109505 428804 057352 574360 090529 724403 882654 523180 633580 247008 195567 094818 103095 138342 528527 872010 448733 788135 060169 368923 090444 355852 596358 125061 513893 892781 945760 180837 578743 311488 141546 881180 725798 154555 747588 968993 208242 843995 928417 863151 333990 452329 646912 066792 331597 065714 130581 029135 411377 168975 575418 044809 948144 743647 568861 209003 932122 360324 630810 447161 706129 092915 663570 598559 623991 959627 374628 765701 988239 791019 099532 481947 145467 960953 831228 883104 232247 660656 297658 372490 343996 402904 930442 720798 139307 478956 422350 342084 323696 026284 054227 819010 359086 880144 085656 781300 908177 717028 043970 924563 694561 / 151 > 32956 [i]
- extracting embedded OOA [i] would yield OOA(32956, 494, S3, 6, 2717), but