Best Known (182, 182+∞, s)-Nets in Base 3
(182, 182+∞, 105)-Net over F3 — Constructive and digital
Digital (182, m, 105)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (182, 104)-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) ≥ 105 from GarcÃa–Stichtenoth tower as constant field extension [i]
(182, 182+∞, 163)-Net over F3 — Digital
Digital (182, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (182, 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
(182, 182+∞, 378)-Net in Base 3 — Upper bound on s
There is no (182, m, 379)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (182, 2267, 379)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32267, 379, S3, 6, 2085), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 58363 877736 405353 551603 104661 562085 288039 214615 935523 028991 193921 791999 334343 684201 034801 891828 468207 472281 354941 262932 454642 256282 962929 720158 646357 285016 768102 717275 619371 947016 674181 133159 195919 637967 466036 112335 148934 034914 319418 574997 648562 914284 444618 095875 386344 704698 356814 748398 944853 503207 904616 617365 814294 489472 420586 686556 634391 341759 975686 645191 978734 924027 384251 209244 582448 232783 406001 974564 427425 773175 019254 296663 046002 415116 750867 131002 152455 684465 386679 779964 321406 812424 291580 116207 785321 013275 959236 126621 180726 302906 842976 906936 479926 918491 951730 648697 095651 893851 332935 667487 539086 686309 419519 897967 788540 970037 555751 344066 603678 325523 383720 534417 306635 562807 960160 007825 393082 470588 928787 550297 815454 073409 167960 873719 195662 679867 404840 380376 280844 964140 903715 057611 480888 747319 300369 548649 492762 125366 957277 326162 025216 189794 325443 974622 484367 069613 117483 924767 077444 515016 951362 802551 814931 904879 684791 028300 342998 061680 529874 901743 102905 251735 022899 929143 984387 988133 151227 606022 531166 291875 277849 076164 952821 946298 091238 950748 841787 390533 664575 576015 203972 / 1043 > 32267 [i]
- extracting embedded OOA [i] would yield OOA(32267, 379, S3, 6, 2085), but