Best Known (80, 80+∞, s)-Nets in Base 7
(80, 80+∞, 61)-Net over F7 — Constructive and digital
Digital (80, m, 61)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (80, 60)-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) ≥ 61 from GarcÃa–Stichtenoth tower as constant field extension [i]
(80, 80+∞, 105)-Net over F7 — Digital
Digital (80, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (80, 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
(80, 80+∞, 503)-Net in Base 7 — Upper bound on s
There is no (80, m, 504)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (80, 1508, 504)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71508, 504, S7, 3, 1428), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 463705 391638 943649 794583 038173 554133 178553 582546 570599 794261 836249 477366 397193 782724 245666 243339 247108 041387 222847 437489 483964 600691 195723 473872 124357 279416 434704 161662 870646 623666 756861 625209 847972 000384 878436 225186 916501 701041 090148 630543 105177 357994 767338 315563 339696 467967 856045 695393 980117 498825 021457 580798 052322 799819 594016 313496 749180 937186 522184 418736 818797 783970 887540 381229 781221 056873 664432 134881 750900 088658 541628 904833 110753 800293 476595 445238 799345 441367 164754 101698 239398 176263 493496 119080 667668 760556 994073 916652 983267 676397 932754 577464 912712 202902 189941 505389 519083 705453 082261 266280 601178 676661 476407 292783 096585 631012 443255 249172 965139 625378 586067 410990 757014 258501 845348 687428 044219 860655 596278 554897 802892 118996 776326 927035 557389 361035 591569 771080 477770 118458 863552 342470 651520 310957 563700 081189 276998 043340 090308 357385 982918 134236 517771 001239 903877 020739 675188 833676 243305 835865 013481 965441 029534 387998 277777 057996 410744 244947 177045 700225 445079 144722 324037 329376 258221 319241 347034 485305 976966 405789 517557 298322 483695 768648 613961 837996 944769 099474 364232 883409 743998 499583 158071 809726 484903 388444 530600 277262 110714 668141 875707 738401 944071 879658 772871 576424 201506 647808 497457 324175 492083 081510 679872 975291 576694 088519 614468 921802 967056 045076 191554 615203 284213 / 1429 > 71508 [i]
- extracting embedded OOA [i] would yield OOA(71508, 504, S7, 3, 1428), but