Best Known (161, ∞, s)-Nets in Base 3
(161, ∞, 84)-Net over F3 — Constructive and digital
Digital (161, m, 84)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (161, 83)-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) ≥ 84 from GarcÃa–Stichtenoth tower as constant field extension [i]
(161, ∞, 163)-Net over F3 — Digital
Digital (161, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (161, 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
(161, ∞, 336)-Net in Base 3 — Upper bound on s
There is no (161, m, 337)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (161, 1678, 337)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31678, 337, S3, 5, 1517), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 119622 465702 200642 863246 820956 989674 079269 479286 156274 434135 024555 302724 072140 514898 981020 314460 898437 830920 660585 561824 804188 487459 826663 416025 017037 726326 480599 506718 883239 852323 555980 833562 619417 884361 721925 844924 275755 047034 892928 514780 208838 701058 433784 070676 710718 168563 053528 412741 249724 029502 116090 457888 980328 459376 022480 931987 778773 436819 432345 657997 864021 763187 527446 782539 733629 688240 753321 151135 005645 004851 233047 995618 957885 399817 169313 756770 262205 925894 709890 409989 534871 767202 768647 309708 870623 294523 004944 707658 207018 935214 642515 279061 655633 081591 293553 305874 544227 356055 203574 983269 483099 405290 792015 935361 160323 339047 205515 232313 688587 099870 867481 144655 819256 390546 960147 528669 367447 596010 030679 524633 081180 465865 042158 703623 804949 972400 185017 997101 347394 520238 836896 260773 578669 086966 / 253 > 31678 [i]
- extracting embedded OOA [i] would yield OOA(31678, 337, S3, 5, 1517), but