Best Known (231, 231+∞, s)-Nets in Base 3
(231, 231+∞, 112)-Net over F3 — Constructive and digital
Digital (231, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (231, 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
(231, 231+∞, 163)-Net over F3 — Digital
Digital (231, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (231, 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
(231, 231+∞, 477)-Net in Base 3 — Upper bound on s
There is no (231, m, 478)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (231, 2383, 478)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32383, 478, S3, 5, 2152), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2 122704 104433 869534 305729 528790 916159 309764 837276 286349 381506 914017 699929 942614 548690 798751 022056 311363 328191 662836 451804 289035 034753 939060 374646 412529 018404 461514 650071 305956 392696 034963 493243 082408 242640 290327 824046 285036 590775 822151 612418 229738 099774 937397 037753 949915 769891 295329 185560 485828 360077 831032 966486 735332 225812 297902 134280 819774 233099 425902 281956 829997 910722 239166 957171 641916 526282 858521 772423 126105 975476 156628 518404 788606 723197 670538 950119 502371 139093 917306 616431 105921 862644 573427 587313 880706 246327 130299 782118 736296 200124 717019 042011 523856 766054 948173 570661 262522 392638 652950 374097 399623 689945 171047 283776 616442 091142 082000 253314 289570 369532 937088 691882 665178 687647 112560 853476 889116 226297 340542 504000 014128 151097 633743 095808 096821 368339 874768 634814 791927 331152 239402 594521 439450 976822 996504 241645 124421 149111 997016 257880 628409 470916 126007 750900 084771 545913 685365 629006 034997 704756 524505 359454 169191 813787 531111 317058 068869 659532 997493 048155 162562 691093 908403 771177 835270 643283 391200 269525 460638 785961 856402 838136 346641 401668 332200 442251 479815 588618 138069 760445 773232 244213 835804 553581 870878 351989 076267 311381 141188 035621 / 2153 > 32383 [i]
- extracting embedded OOA [i] would yield OOA(32383, 478, S3, 5, 2152), but