Best Known (233, ∞, s)-Nets in Base 3
(233, ∞, 112)-Net over F3 — Constructive and digital
Digital (233, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (233, 111)-sequence over F3, using
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on function field F4/F3 with g(F4) = 79, N(F4) ≥ 2, and N(F4) + N2(F4) ≥ 112 from GarcÃa–Stichtenoth tower as constant field extension [i]
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
(233, ∞, 163)-Net over F3 — Digital
Digital (233, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (233, 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
(233, ∞, 481)-Net in Base 3 — Upper bound on s
There is no (233, m, 482)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (233, 2403, 482)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32403, 482, S3, 5, 2170), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 7341 481101 308823 283118 041542 707471 576603 761723 037847 569663 988683 762300 674645 999199 251130 988004 158884 956626 261630 410995 682262 799604 688337 082225 238907 444186 726586 995673 044100 286963 026305 409264 845708 167382 179096 783182 894374 676348 929888 687113 955611 044598 986255 728955 393116 426713 447830 505160 531264 878280 701555 206303 681629 999214 680800 892251 314038 813200 958056 473450 845982 814050 090603 530075 658605 052448 942747 019491 785173 599607 024178 593275 011181 327198 856250 227989 442111 330464 472840 785344 356223 695484 621542 479674 431298 221222 940857 735756 326318 095622 113591 908303 853876 665071 876442 923542 742629 746643 395080 253316 487175 013455 934681 471545 534020 290755 712298 742189 485452 586917 256746 642342 009891 520249 884910 324370 005196 672849 432147 704008 166229 532895 268174 449563 715605 340607 201145 341475 677283 781329 359763 067895 195908 375483 148607 301706 429740 203623 016844 204738 452567 903903 055051 737158 622104 357473 544214 126306 939068 872024 165522 353538 819818 939455 940723 687345 812132 541492 217206 347633 365481 172253 320855 282818 126923 512402 916486 419322 711947 764276 538088 170738 488079 506842 830321 629903 124297 487203 983147 599148 656565 115709 540004 744796 416756 410863 569247 451655 280742 310547 355174 539535 / 2171 > 32403 [i]
- extracting embedded OOA [i] would yield OOA(32403, 482, S3, 5, 2170), but