Best Known (243, ∞, s)-Nets in Base 3
(243, ∞, 116)-Net over F3 — Constructive and digital
Digital (243, m, 116)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (243, 115)-sequence over F3, using
- base reduction for sequences [i] based on digital (64, 115)-sequence over F9, using
- s-reduction based on digital (64, 164)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- T4 from the second 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) = 64 and N(F) ≥ 165, using
- s-reduction based on digital (64, 164)-sequence over F9, using
- base reduction for sequences [i] based on digital (64, 115)-sequence over F9, using
(243, ∞, 163)-Net over F3 — Digital
Digital (243, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (243, 162)-sequence over F3, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 151 and N(F) ≥ 163, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
(243, ∞, 501)-Net in Base 3 — Upper bound on s
There is no (243, m, 502)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (243, 3004, 502)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(33004, 502, S3, 6, 2761), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 402 614663 604016 064490 593149 304764 611675 922538 777918 924940 760307 578058 576527 013659 725904 801662 904115 864348 355372 172739 625375 321872 439086 919739 044346 818413 103736 181235 055206 677964 179436 657445 348279 906732 705261 702068 534473 701867 574841 205119 702864 870348 426462 501264 678131 414422 986636 607244 504536 959942 324676 439528 297026 880580 520646 501523 261931 621687 367016 847534 462668 383045 605147 811135 725789 010122 094416 947255 201855 570835 869917 197171 081716 033984 740800 635357 679985 303186 399561 045378 349627 623001 056329 849010 718945 483989 684752 807837 159340 385130 254491 685796 542807 475483 025279 785126 933553 725050 894140 634031 223423 029118 959952 209080 496853 782559 686347 670357 427039 778921 824987 211661 072565 976302 174128 916208 595912 767952 426649 869651 647955 236837 012721 698300 345521 241686 334902 676536 768116 453841 350659 719975 952257 099155 124115 854174 556477 676080 706687 924893 457026 873229 593759 179811 479808 341727 840666 874073 986301 626796 379931 562071 952267 634982 355533 377994 201424 575559 848872 463987 393717 193672 188481 635298 873110 477936 883825 083798 845885 197816 136720 444828 519002 938793 922838 659590 144281 052628 073882 836896 425743 201210 428360 714794 378162 862553 270840 402548 317688 691913 415449 462212 254561 346173 629670 105436 984857 693776 360071 920597 992404 118143 639329 296285 057754 017488 096688 838143 884162 184280 100850 721716 362706 967330 396179 382039 871663 371421 910709 396863 826518 881089 126116 438842 585510 051647 495522 539328 611227 511971 186808 750451 664326 420165 910082 174039 368553 754398 044707 634231 / 1381 > 33004 [i]
- extracting embedded OOA [i] would yield OOA(33004, 502, S3, 6, 2761), but