Best Known (209, ∞, s)-Nets in Base 3
(209, ∞, 112)-Net over F3 — Constructive and digital
Digital (209, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (209, 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
(209, ∞, 163)-Net over F3 — Digital
Digital (209, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (209, 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
(209, ∞, 432)-Net in Base 3 — Upper bound on s
There is no (209, m, 433)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (209, 2591, 433)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32591, 433, S3, 6, 2382), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 4378 161995 079049 499322 177159 706567 173471 944659 050050 851053 310192 080524 358134 876788 671947 610433 891177 067967 838358 343672 141900 947988 595963 911708 907449 448737 654127 605181 508882 668738 923738 598714 642649 561053 202551 596153 919090 350023 002606 384317 630652 552618 592953 821908 381591 263355 193196 124463 337598 640223 955744 490635 442839 414919 821313 967751 679973 937636 475963 761658 902842 429826 821782 603222 239676 215380 392353 937810 818409 473894 905290 972367 269514 475757 228876 867811 847939 641328 209941 129474 172785 724030 225969 868032 562197 831959 578837 299013 396075 110585 392529 223595 368881 039411 433208 243532 611435 563337 156448 243269 130565 182038 450885 881884 443577 260656 865744 606442 598052 421724 353015 492654 653430 795508 077448 545104 535395 048454 738080 496124 956086 847981 376747 846026 797347 911355 297246 245186 706911 141651 359347 213263 966182 534657 117069 198490 378237 131127 891937 653398 390425 213173 427865 070483 603114 279857 895395 759909 086599 611144 339608 112634 659324 295956 236304 739628 413578 373362 339679 170392 928876 644840 783079 141055 526941 657660 002569 732567 576038 253445 248548 564118 027539 128879 580524 898702 959608 761342 887006 214414 316316 374201 344737 891865 256735 049115 000776 849648 201153 233147 236752 445335 632771 788534 840822 022640 495066 690661 733727 034107 764879 203310 854415 835218 860570 459611 006957 / 2383 > 32591 [i]
- extracting embedded OOA [i] would yield OOA(32591, 433, S3, 6, 2382), but