Best Known (122, ∞, s)-Nets in Base 3
(122, ∞, 78)-Net over F3 — Constructive and digital
Digital (122, m, 78)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (122, 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
(122, ∞, 120)-Net over F3 — Digital
Digital (122, m, 120)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (122, 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
(122, ∞, 257)-Net in Base 3 — Upper bound on s
There is no (122, m, 258)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (122, 1541, 258)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31541, 258, S3, 6, 1419), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 157796 015419 145208 641492 371021 758380 475525 067416 748260 000880 164362 097792 375793 976320 401537 519992 642870 225066 719349 202183 195942 322171 894883 571808 139589 671157 219874 540351 745779 400234 380758 778526 370585 624136 091075 443895 966413 096036 758359 326636 657175 728708 296530 788546 731483 063433 955926 060715 221158 540115 241525 402027 610286 750461 677383 040410 307077 991973 949019 932578 173269 658181 989630 722719 016666 472376 316384 026627 467411 035141 690559 225893 130809 314920 246029 856791 334067 588580 630204 756646 505779 754785 343361 396897 512745 296329 032962 470364 191608 617714 501369 599030 826921 658330 924134 185154 452434 689664 957099 813369 691368 233132 287735 885964 655146 972155 742690 185913 008829 187347 513876 518919 605856 923442 351706 138064 436121 085305 014689 582175 493825 978314 876270 / 71 > 31541 [i]
- extracting embedded OOA [i] would yield OOA(31541, 258, S3, 6, 1419), but