Best Known (177, ∞, s)-Nets in Base 3
(177, ∞, 100)-Net over F3 — Constructive and digital
Digital (177, m, 100)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (177, 99)-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) ≥ 100 from GarcÃa–Stichtenoth tower as constant field extension [i]
(177, ∞, 163)-Net over F3 — Digital
Digital (177, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (177, 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
(177, ∞, 368)-Net in Base 3 — Upper bound on s
There is no (177, m, 369)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (177, 2207, 369)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32207, 369, S3, 6, 2030), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 922939 109356 566530 635996 838776 977459 421957 718363 617636 698805 236930 387677 088791 080357 361556 955516 828555 937996 514169 948477 311568 947549 445892 394772 836228 710819 994535 227771 120807 761651 468680 184201 073676 830720 175325 777213 409367 695949 045401 818708 593336 592972 503366 469166 102985 972366 888997 410513 646432 636028 099950 545450 218789 079095 851094 553682 969608 347596 731164 832004 915223 494425 626674 796410 633568 719722 823751 325609 307640 695272 268159 812944 790760 002811 320339 100992 135769 745086 637209 571970 685821 255296 844423 035185 384029 548023 042730 608752 214981 039395 487197 918126 668269 565139 718338 860208 539727 330637 616818 558266 827356 898259 079813 590030 449231 151529 267149 636865 510599 460200 076962 704873 810380 709815 757536 952515 283640 004446 885952 320846 530721 233502 956993 104733 686820 587484 948809 325140 415483 097268 574351 639337 826206 357100 667462 738984 015316 525570 268712 518847 394520 344874 403983 605249 405401 516488 215684 045055 979325 152152 083812 684472 087735 620527 909533 684059 056289 406216 361250 058082 662596 822980 890130 946896 361501 877437 843517 313728 743775 601017 883570 672113 276215 209059 345312 759983 / 677 > 32207 [i]
- extracting embedded OOA [i] would yield OOA(32207, 369, S3, 6, 2030), but