Best Known (234, 234+∞, s)-Nets in Base 3
(234, 234+∞, 112)-Net over F3 — Constructive and digital
Digital (234, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (234, 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
(234, 234+∞, 163)-Net over F3 — Digital
Digital (234, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (234, 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
(234, 234+∞, 483)-Net in Base 3 — Upper bound on s
There is no (234, m, 484)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (234, 2413, 484)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32413, 484, S3, 5, 2179), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 107 934425 186213 334974 428862 605613 644231 589416 673483 157263 699677 285862 390386 805991 876372 987985 102499 011698 094985 730050 905822 686119 531803 014687 622065 579534 606951 479044 723887 533667 500670 035897 193764 602106 615379 194461 061920 513249 336349 472743 709836 152942 547175 119538 303295 902223 433245 867565 112966 735704 707860 874024 484048 533711 188934 217653 385381 238166 215941 668892 851997 023634 275395 113953 851801 890425 573529 944465 274268 512934 183287 404191 885976 530683 982371 534295 777398 372037 177126 789010 456050 325847 652883 531325 796020 887545 161630 682509 221200 319146 660413 529813 059144 168795 696257 782733 586938 702769 315493 884598 217144 175671 604606 968893 921740 980461 546438 315241 144336 799502 438786 577883 685117 825090 340255 911242 315063 870850 776543 886841 684370 077342 709771 903744 528809 834960 034252 433907 741360 586653 986772 422910 876525 774748 972126 403534 964249 606066 562721 712817 067975 618822 905556 582659 903053 367561 109281 837945 315123 060192 952865 084476 330367 595109 844752 944383 098181 622371 416367 195407 777789 780253 728844 629384 909315 016716 394566 334698 162432 187818 361972 047640 939411 995978 666528 764805 604590 959925 046481 852397 999470 662549 263225 682247 773471 004112 694109 918608 736134 798834 392478 848099 939327 / 545 > 32413 [i]
- extracting embedded OOA [i] would yield OOA(32413, 484, S3, 5, 2179), but