Best Known (86, ∞, s)-Nets in Base 7
(86, ∞, 67)-Net over F7 — Constructive and digital
Digital (86, m, 67)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (86, 66)-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) ≥ 67 from GarcÃa–Stichtenoth tower as constant field extension [i]
(86, ∞, 105)-Net over F7 — Digital
Digital (86, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (86, 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
(86, ∞, 539)-Net in Base 7 — Upper bound on s
There is no (86, m, 540)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (86, 1616, 540)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71616, 540, S7, 3, 1530), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 8 446002 861529 668531 773783 317114 294261 407213 062921 188564 143273 568736 389330 715879 740472 145028 115379 079727 667635 909819 648788 098431 636378 830321 298959 006169 867646 316902 965814 300773 928694 514530 538534 299180 060113 835747 223030 376497 477612 371209 866614 314182 676481 810190 885366 193247 482427 998914 054437 474370 664635 110310 394661 487102 160521 580794 388194 539411 679850 983577 707982 996521 479462 028701 731681 673522 109621 124138 681769 119933 936875 797567 224767 311747 453683 499788 456907 379546 200699 313654 363442 388236 330325 457941 892886 150056 809475 097479 002627 800225 433394 433142 640731 922487 891342 250213 609845 853110 219548 400869 770468 272961 546875 189274 006158 764916 212603 278955 513182 179040 170020 170437 365944 538829 016854 238090 052278 282368 911415 666957 423523 444799 724143 221938 234986 853314 419968 425771 511369 707740 135628 255184 905743 611867 526133 611397 998312 400138 608730 396587 012963 469632 271298 633139 714410 899629 979643 650586 268167 017459 220576 402556 256673 687556 741342 735452 391071 576424 134721 464452 534409 712253 621309 331793 988911 965035 631792 029178 889007 208752 438837 695125 660348 661515 165861 850105 458833 484264 479954 488132 289111 459725 023443 340367 330367 554408 452712 329674 341026 428189 865243 522158 248712 285689 980157 944357 935738 274642 850314 738820 163104 709274 648522 587175 829738 973754 235215 362349 085691 090756 673429 546106 067190 901621 041592 320286 349640 028908 467081 283871 142153 682463 002287 534866 000898 396860 165702 033482 923371 / 1531 > 71616 [i]
- extracting embedded OOA [i] would yield OOA(71616, 540, S7, 3, 1530), but