Best Known (207, ∞, s)-Nets in Base 3
(207, ∞, 112)-Net over F3 — Constructive and digital
Digital (207, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (207, 111)-sequence over F3, using
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on function field F4/F3 with g(F4) = 79, N(F4) ≥ 2, and N(F4) + N2(F4) ≥ 112 from GarcÃa–Stichtenoth tower as constant field extension [i]
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
(207, ∞, 163)-Net over F3 — Digital
Digital (207, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (207, 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
(207, ∞, 428)-Net in Base 3 — Upper bound on s
There is no (207, m, 429)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (207, 2567, 429)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32567, 429, S3, 6, 2360), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 5179 045666 484127 700593 896479 496996 405561 179141 909694 024548 682064 496163 718542 665272 369655 552820 311247 785471 104122 793830 864490 867499 500136 235553 665914 264712 707269 620002 500438 172544 957287 292043 894000 657376 739509 600068 392922 699168 825875 071982 827362 216876 309888 685135 304846 863790 167957 135325 278916 762394 003201 425726 607790 652605 544635 568283 303863 970241 778456 698380 442693 781028 108324 118916 105491 524781 482579 829673 934183 267123 158789 567504 448497 987033 995218 219239 237520 273265 267504 054990 809349 080189 496201 873161 565048 843502 041227 603497 864568 533388 090388 841679 898367 989498 497960 116756 251527 298950 745650 729322 994962 720330 293158 575439 787163 581647 584735 950260 426132 711194 958875 145842 302485 716465 240335 087065 167368 668642 280111 615924 997610 317677 117258 083648 364640 813222 823982 498527 520751 714159 343414 953199 386257 301426 546293 546497 539947 463085 997629 892656 555790 419086 793464 650495 466960 975392 143578 973666 015086 839013 893330 743616 624187 882063 293131 249341 381705 725124 645133 808456 868953 575635 642712 262601 068633 082972 234921 291667 224802 116178 592387 789375 578865 480849 222271 088562 369164 920569 362653 638279 874783 715327 702237 479741 136562 103693 648961 928080 120658 464963 033974 289029 054276 465110 729788 480378 282509 682222 462132 869393 740627 333006 480116 412758 555973 / 787 > 32567 [i]
- extracting embedded OOA [i] would yield OOA(32567, 429, S3, 6, 2360), but