Best Known (140, 140+∞, s)-Nets in Base 3
(140, 140+∞, 78)-Net over F3 — Constructive and digital
Digital (140, m, 78)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (140, 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
(140, 140+∞, 120)-Net over F3 — Digital
Digital (140, m, 120)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (140, 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
(140, 140+∞, 293)-Net in Base 3 — Upper bound on s
There is no (140, m, 294)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (140, 1757, 294)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31757, 294, S3, 6, 1617), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 17 500835 641109 204207 170290 793394 770544 048937 117246 786972 317846 384497 895956 677310 542159 535775 315454 682116 097615 235223 710262 188475 681564 778375 050732 986094 638261 533344 582818 475724 390349 424277 829033 199168 076506 357403 357410 058753 416434 571606 998295 671559 412048 545939 245725 529626 437819 340419 873480 655950 250400 495969 258048 197067 690137 847506 104946 048458 215221 762460 151036 377507 371124 040081 322792 229105 875104 458903 601105 633730 737333 403812 562319 461959 723533 709749 555142 454935 489147 957473 595631 868074 014701 825249 064772 660722 053233 202578 599227 713476 784152 137911 303744 715897 195906 619613 596853 685306 154500 956719 749065 058468 219863 539242 261121 897697 709967 784355 263016 386644 562728 526772 253787 099366 166611 220622 839111 299819 407859 068924 262553 301147 816237 204796 684937 008809 037815 698244 657419 071617 903008 088840 716783 049875 969240 512938 847002 443902 285098 141673 799499 / 809 > 31757 [i]
- extracting embedded OOA [i] would yield OOA(31757, 294, S3, 6, 1617), but