Best Known (121, ∞, s)-Nets in Base 3
(121, ∞, 78)-Net over F3 — Constructive and digital
Digital (121, m, 78)-net over F3 for arbitrarily large m, using
- net from sequence [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
(121, ∞, 120)-Net over F3 — Digital
Digital (121, m, 120)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (121, 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
(121, ∞, 255)-Net in Base 3 — Upper bound on s
There is no (121, m, 256)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (121, 1529, 256)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31529, 256, S3, 6, 1408), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 5 948318 205715 295482 379474 860745 080304 667910 173626 404703 471566 224762 533891 680178 970544 947545 103192 914171 423550 831849 663593 936820 558024 241325 418794 804653 283273 787344 142397 695662 644323 818700 846337 205319 129548 450115 928922 546702 940250 612576 956402 872103 893231 560668 943280 980160 301507 756681 832341 872775 002885 874240 453069 659180 544185 093435 099825 728781 004623 489780 896310 354618 516284 075940 970187 911517 923166 519381 580966 210362 148708 638468 789685 127568 144414 140920 248380 771940 236511 594768 359018 214855 623734 422711 052616 387013 365155 543164 383182 025021 231357 467911 396719 923122 018366 755334 099666 498773 993013 913553 767660 913899 386291 543762 930394 765886 853893 555873 040388 823741 587862 424351 142816 300085 427410 466724 056337 775267 136478 231101 156257 031317 554049 / 1409 > 31529 [i]
- extracting embedded OOA [i] would yield OOA(31529, 256, S3, 6, 1408), but