Best Known (138, ∞, s)-Nets in Base 3
(138, ∞, 78)-Net over F3 — Constructive and digital
Digital (138, m, 78)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (138, 77)-sequence over F3, using
- t-expansion [i] based on digital (121, 77)-sequence over F3, using
- base reduction for sequences [i] based on digital (22, 77)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 22 and N(F) ≥ 78, using
- F3 from the tower of function fields by GarcÃa and Stichtenoth over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 22 and N(F) ≥ 78, using
- base reduction for sequences [i] based on digital (22, 77)-sequence over F9, using
- t-expansion [i] based on digital (121, 77)-sequence over F3, using
(138, ∞, 120)-Net over F3 — Digital
Digital (138, m, 120)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (138, 119)-sequence over F3, using
- t-expansion [i] based on digital (113, 119)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 113 and N(F) ≥ 120, using
- t-expansion [i] based on digital (113, 119)-sequence over F3, using
(138, ∞, 289)-Net in Base 3 — Upper bound on s
There is no (138, m, 290)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (138, 1733, 290)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31733, 290, S3, 6, 1595), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 10 363042 636273 157983 746533 305893 585951 474803 079581 214903 615291 573300 560396 987258 425165 133821 644893 170340 840965 781953 129243 796857 783121 532633 633197 210031 386941 653475 240861 857865 659562 725746 523085 330910 942715 429864 450081 257371 589746 656522 799917 664308 432271 817731 150331 854116 151082 307644 212360 995994 079952 564770 834032 682245 741140 057391 294333 675287 328651 802225 760968 805932 131735 812079 613999 260242 704013 460118 373255 494420 957199 884962 139549 631557 736131 519253 884898 051667 521038 464333 267921 569824 048802 142366 816700 749305 981902 425821 319088 952354 008041 993322 870740 095319 877163 791502 380047 861534 739429 636628 374843 368867 070429 184407 646225 789722 750751 164887 203506 114515 533491 827451 716705 107656 616074 591341 635623 950525 500666 396528 343625 681905 696997 288697 384880 664363 234655 764121 242427 392729 621021 323507 878826 001953 413344 358886 199235 550731 998358 / 133 > 31733 [i]
- extracting embedded OOA [i] would yield OOA(31733, 290, S3, 6, 1595), but