Best Known (91, ∞, s)-Nets in Base 7
(91, ∞, 72)-Net over F7 — Constructive and digital
Digital (91, m, 72)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (91, 71)-sequence over F7, using
- Niederreiter–Xing sequence construction III based on function field F2/F7 with g(F2) = 21, N(F2) ≥ 2, and N(F2) + N2(F2) ≥ 72 from GarcÃa–Stichtenoth tower as constant field extension [i]
(91, ∞, 105)-Net over F7 — Digital
Digital (91, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (91, 104)-sequence over F7, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 43 and N(F) ≥ 105, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
(91, ∞, 569)-Net in Base 7 — Upper bound on s
There is no (91, m, 570)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (91, 1706, 570)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71706, 570, S7, 3, 1615), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 11849 935819 577169 034294 880721 303355 022006 006855 956056 286526 916354 106127 600430 737856 192085 486286 284366 037311 216977 156789 954922 148119 128636 088892 054417 518044 893761 334128 424352 905636 079333 385780 856158 733295 759199 543454 810166 568648 998584 519099 613839 238310 063459 488024 989096 363894 816542 123480 231518 753704 785730 104980 613122 145921 220544 191175 638771 484409 143286 400949 298553 536081 387723 134057 279104 140095 123518 546196 642719 184629 873387 593089 635869 975651 028315 161989 035800 991017 312357 447600 278764 845827 400191 430667 288067 196359 436214 867386 217829 852041 831187 042214 019053 555656 083723 741885 538213 459669 230685 796237 260930 023737 104127 079725 612819 324506 844065 824218 864842 523445 510303 662846 697805 252674 538627 632374 895173 090350 603542 517179 270305 281671 792669 189069 737871 878812 517700 795208 739928 894747 849880 406743 954665 730799 675365 293079 579955 924200 623774 334031 644636 747958 948300 722445 956384 456693 659809 015213 519060 042571 833658 572722 937116 039141 212985 408937 015412 006712 068316 381882 899748 535660 100691 712857 005671 706869 884852 886255 538142 102360 816894 581811 822006 513113 718579 313051 938531 354125 523849 521877 552334 273905 039956 281914 707850 032231 627272 379717 478518 315119 010408 016566 800101 167712 141452 117530 381186 932442 590467 717779 190011 282159 853248 818729 378496 239911 106302 652305 128045 953857 180682 577351 398284 575927 968343 519277 442214 108992 989343 646051 153753 508357 306426 133958 420906 825136 412172 265024 878609 129446 452660 424125 703617 439067 392210 541373 379730 421777 672427 998663 189833 / 202 > 71706 [i]
- extracting embedded OOA [i] would yield OOA(71706, 570, S7, 3, 1615), but