Best Known (180, ∞, s)-Nets in Base 3
(180, ∞, 103)-Net over F3 — Constructive and digital
Digital (180, m, 103)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (180, 102)-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) ≥ 103 from GarcÃa–Stichtenoth tower as constant field extension [i]
(180, ∞, 163)-Net over F3 — Digital
Digital (180, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (180, 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
(180, ∞, 374)-Net in Base 3 — Upper bound on s
There is no (180, m, 375)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (180, 2243, 375)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32243, 375, S3, 6, 2063), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 69035 510105 781740 644065 541853 790532 831210 456621 985294 171179 854096 938715 986366 130529 361693 013960 595527 820444 727849 112709 643379 848812 955411 154695 739487 906461 731806 649708 863626 150662 161402 959680 224400 575404 727681 256418 204007 915875 346071 366691 225380 322141 244433 768517 423435 423121 935432 450302 442942 433093 290597 146901 315956 784925 133777 717211 782976 304706 172927 028681 554808 779784 392515 485747 068720 428001 641134 459648 973787 619716 160392 879503 813477 016252 138758 542165 861914 573221 232279 626681 728473 858644 921460 090241 586281 703827 327463 397987 592434 449624 601019 623675 999165 757722 331081 169136 183990 498792 847266 630337 757095 827961 995095 104050 877947 257792 729999 002010 313355 279569 311538 010860 370928 129446 849311 793569 145189 232308 653183 188479 532587 029811 899033 924825 548183 550668 645307 781710 615699 248574 617917 698936 019966 834626 777431 237141 359407 520032 369227 363615 856137 369181 358494 654520 463860 469091 255514 620494 537737 514564 142307 446248 508871 813210 581788 928134 218691 178663 656993 133814 897169 404346 960890 229459 838119 272524 411531 608378 358127 470938 275961 257917 282405 816455 862642 833743 300802 557410 169031 / 344 > 32243 [i]
- extracting embedded OOA [i] would yield OOA(32243, 375, S3, 6, 2063), but