Best Known (124, ∞, s)-Nets in Base 3
(124, ∞, 78)-Net over F3 — Constructive and digital
Digital (124, m, 78)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (124, 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
(124, ∞, 120)-Net over F3 — Digital
Digital (124, m, 120)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (124, 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
(124, ∞, 261)-Net in Base 3 — Upper bound on s
There is no (124, m, 262)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (124, 1565, 262)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31565, 262, S3, 6, 1441), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 444177 013083 999838 424825 636316 003194 440688 525384 231255 045508 179479 049932 970039 056015 344865 316193 482217 132755 795941 175049 552592 931268 489777 020657 637615 625245 672612 962224 081028 712390 867593 270113 269987 972804 206942 058176 046511 201369 703688 905492 875816 348804 421020 695439 001337 146114 362453 789992 051478 973956 004339 815158 516670 148702 526318 590446 262750 251728 962643 935933 424033 544403 454384 122080 026167 388539 930388 530369 000226 197162 114932 448996 661906 909909 445028 826176 164130 111782 622880 995401 680512 399500 600393 210503 790854 621125 816978 852969 765774 978969 688415 832009 614700 663848 410794 752017 872370 810214 522518 394099 200280 246965 391720 947226 342686 193290 207645 937396 941367 507249 187131 662440 876620 438536 260016 381892 177465 747937 650060 859779 578665 987716 377061 949563 185171 / 721 > 31565 [i]
- extracting embedded OOA [i] would yield OOA(31565, 262, S3, 6, 1441), but