Best Known (160, ∞, s)-Nets in Base 3
(160, ∞, 83)-Net over F3 — Constructive and digital
Digital (160, m, 83)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (160, 82)-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) ≥ 83 from GarcÃa–Stichtenoth tower as constant field extension [i]
(160, ∞, 163)-Net over F3 — Digital
Digital (160, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (160, 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
(160, ∞, 334)-Net in Base 3 — Upper bound on s
There is no (160, m, 335)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (160, 1668, 335)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31668, 335, S3, 5, 1508), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 4 072305 522693 454285 123220 688823 803854 636133 639853 719906 698829 480841 859915 409526 960676 944018 811909 754688 805901 797807 438285 675772 564809 691552 079529 998854 649997 818847 581724 758899 804717 792007 435512 054036 619752 882402 309614 589153 778870 247663 115430 792555 907133 414183 157348 741408 330010 521372 134388 941531 749646 327940 752790 422880 633738 017776 243526 634981 990644 948988 167485 117390 805481 664486 906641 617433 702315 787591 622038 582288 793653 715896 126553 959064 741455 186268 699662 033462 938814 538394 568872 128290 917668 415736 362268 401923 725925 299346 237985 468092 780310 191278 356360 932984 586907 385115 519294 631609 849943 907579 990483 201357 648367 099330 384234 432713 922525 686294 565771 673483 674499 472037 491035 012690 763871 158726 900948 984806 532861 508630 561874 588512 234462 714512 197564 315341 109448 093877 314096 013086 414059 245255 561257 322751 / 503 > 31668 [i]
- extracting embedded OOA [i] would yield OOA(31668, 335, S3, 5, 1508), but