Best Known (89, 89+∞, s)-Nets in Base 7
(89, 89+∞, 70)-Net over F7 — Constructive and digital
Digital (89, m, 70)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (89, 69)-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) ≥ 70 from GarcÃa–Stichtenoth tower as constant field extension [i]
(89, 89+∞, 105)-Net over F7 — Digital
Digital (89, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (89, 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
(89, 89+∞, 557)-Net in Base 7 — Upper bound on s
There is no (89, m, 558)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (89, 1670, 558)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71670, 558, S7, 3, 1581), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2574 167701 344102 904008 395770 141152 989140 687403 831492 519799 059076 775384 809213 096702 294646 262228 254222 502626 470306 806856 886943 973494 871609 380454 149242 817973 652103 915726 407294 987186 617485 524051 629360 595994 213449 711225 947278 493099 444343 688325 305224 648763 598302 736015 847718 392533 501097 366095 648488 438457 101048 209448 257126 281601 760454 574017 853386 991985 974957 064287 130568 271084 651130 878437 974387 273217 756981 146337 015407 027688 943716 035799 577787 705583 341842 504475 805256 597417 074679 111827 354971 512569 590299 303035 252742 811501 845447 354710 263619 602165 840405 275657 701286 984310 945162 261158 440043 935616 537036 056646 908185 166483 577900 360355 255093 789332 085235 957675 252703 315622 153400 070073 542224 761892 158346 391395 551083 380128 982292 013730 046879 335839 823083 136911 554629 162742 780560 469044 625895 277130 249373 298031 321535 854020 699147 990371 537286 558400 067718 053890 959083 969580 911059 764629 889249 729886 444822 537336 682824 470004 264122 955167 410097 178968 019581 457895 784403 779429 429543 713592 415465 892867 444668 947273 064032 266952 425289 506130 092517 562826 321781 531686 226686 316728 376463 894122 434260 293450 953080 620675 371737 353630 637101 777800 781751 435328 089122 166495 090476 650881 563444 149431 764491 005266 227481 420679 988336 604487 217727 972695 676036 220530 153723 536793 745409 560078 644535 892390 957471 819775 950558 416740 842629 869171 463441 927551 829940 983342 398814 762830 699777 164694 508015 671312 032066 851573 998871 352576 292023 193922 670285 060807 232650 253622 612209 906125 / 113 > 71670 [i]
- extracting embedded OOA [i] would yield OOA(71670, 558, S7, 3, 1581), but