Best Known (227, 227+∞, s)-Nets in Base 3
(227, 227+∞, 112)-Net over F3 — Constructive and digital
Digital (227, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (227, 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
(227, 227+∞, 163)-Net over F3 — Digital
Digital (227, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (227, 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
(227, 227+∞, 469)-Net in Base 3 — Upper bound on s
There is no (227, m, 470)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (227, 2343, 470)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32343, 470, S3, 5, 2116), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 177425 500534 523129 915160 602798 165800 077659 052010 456435 162973 496274 226998 703461 783957 362039 907329 587782 361518 381493 301198 229613 603814 411077 937159 881040 010078 453403 436753 254838 509416 419904 755400 802641 047471 573053 600740 267731 993619 953338 158357 787920 335427 030656 626184 924102 143216 651721 979825 090727 458605 176351 130283 268647 753173 487874 302314 977276 510482 013902 450703 941408 127209 643796 965680 547291 046474 315819 255478 326782 610272 805707 716334 875062 509506 591766 083666 852470 779967 956350 584455 036973 573040 221562 541132 911522 146757 075573 182992 342532 617916 872124 701329 407150 261258 892955 963395 124291 900973 843194 054611 083977 488457 765644 063070 135456 956905 353498 001226 246229 522978 257472 601403 330019 228841 291860 193753 621326 236700 646286 293839 981425 391397 727078 716024 003857 305557 072752 780856 985888 973086 121015 786117 199276 826606 230420 996904 365443 759040 684094 358478 246125 319488 060012 237594 362354 218938 555795 754553 079157 292459 096891 606350 294044 315926 271564 583992 835421 041314 662509 520326 283841 078709 068324 325314 429792 287750 716901 882087 623535 235069 267036 791497 769853 541415 637886 633814 251523 989944 055249 292229 494305 122341 400163 379528 772828 721939 717393 / 2117 > 32343 [i]
- extracting embedded OOA [i] would yield OOA(32343, 470, S3, 5, 2116), but