Best Known (169, 169+∞, s)-Nets in Base 3
(169, 169+∞, 92)-Net over F3 — Constructive and digital
Digital (169, m, 92)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (169, 91)-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) ≥ 92 from GarcÃa–Stichtenoth tower as constant field extension [i]
(169, 169+∞, 163)-Net over F3 — Digital
Digital (169, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (169, 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
(169, 169+∞, 352)-Net in Base 3 — Upper bound on s
There is no (169, m, 353)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (169, 1758, 353)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31758, 353, S3, 5, 1589), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 16 959572 683136 754592 515539 531743 385888 047423 598156 680158 741005 774668 064122 965641 143948 416112 161368 454834 156451 877433 286233 460997 052238 032652 111019 594772 123676 125096 812215 842454 563843 771980 576588 873420 610222 655627 995840 675493 001493 296196 472575 186872 007552 199157 619569 069947 269639 360819 258836 924322 923068 521867 116046 706436 730649 048098 699638 438712 084854 079085 094818 757584 462738 760491 178788 345731 466596 073576 685607 521347 312467 422251 349051 849734 165073 904293 383333 925401 401854 721675 443189 645350 076102 799725 897820 928947 144370 320024 621932 010998 120724 752202 706721 683446 767167 239625 547466 457925 551784 432285 117650 675216 625434 769987 345829 467872 007597 646694 790964 333449 370066 819758 678927 704540 408880 976686 050272 806010 560193 324524 543092 889772 110580 384029 777155 451835 562419 336443 276261 780743 122502 684031 210078 419467 434109 363054 140394 120894 997930 157910 692298 / 265 > 31758 [i]
- extracting embedded OOA [i] would yield OOA(31758, 353, S3, 5, 1589), but