Best Known (212, 212+∞, s)-Nets in Base 4
(212, 212+∞, 200)-Net over F4 — Constructive and digital
Digital (212, m, 200)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (212, 199)-sequence over F4, using
- t-expansion [i] based on digital (161, 199)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 161 and N(F) ≥ 200, using
- F7 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 161 and N(F) ≥ 200, using
- t-expansion [i] based on digital (161, 199)-sequence over F4, using
(212, 212+∞, 258)-Net over F4 — Digital
Digital (212, m, 258)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (212, 257)-sequence over F4, using
- t-expansion [i] based on digital (191, 257)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 191 and N(F) ≥ 258, using
- t-expansion [i] based on digital (191, 257)-sequence over F4, using
(212, 212+∞, 653)-Net in Base 4 — Upper bound on s
There is no (212, m, 654)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (212, 3264, 654)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(43264, 654, S4, 5, 3052), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 4 287525 614354 115710 996493 211612 430955 407666 816243 887398 428990 601638 145248 992312 809394 397532 523937 638658 499311 659747 656612 320799 073918 020170 693469 390515 460804 211059 347381 630041 956287 458674 034362 046279 519948 444656 779971 124812 429607 720087 935027 857718 656587 667349 050449 225902 062732 074521 949038 989549 519928 530960 109961 122646 324613 038745 909521 094441 001922 304429 370223 676245 860293 950253 135448 686222 576581 579576 162366 972540 485762 945456 199207 118419 908454 103135 853890 328789 832510 990626 459860 669997 966773 026243 073843 986932 904882 299873 910843 104591 641007 715062 861596 744898 874042 203036 846776 454112 182531 902217 530710 801781 429614 152562 099423 751914 439649 739778 729363 797848 038257 748233 542026 005572 535634 014766 687490 984871 734327 709681 376214 986724 455196 796912 215314 950207 312504 180494 883396 014011 792724 265924 052140 608962 470148 434130 763065 826501 201143 148331 211451 130422 873037 770290 243734 059391 280689 583118 413668 393216 057961 209338 250001 464498 306798 294660 109188 372447 405651 151127 021351 476632 850842 701594 999139 470040 546136 959706 895924 079995 876479 817766 845407 970675 555453 620136 896927 907048 170992 968037 152203 818881 645802 358692 629660 063724 071287 594796 185157 487666 119524 556291 030751 510110 702122 863705 123412 976635 072110 681961 362181 387411 449410 946169 602804 016698 978218 283981 408498 926506 005569 637742 952063 628963 536700 857499 107168 243691 577812 960425 877476 258275 828450 333058 764169 230393 657341 296838 230897 537029 831983 739312 311815 431875 522904 019146 043974 336995 882859 158261 047754 324999 223326 208514 519559 730222 418535 304483 195175 573838 807597 642090 958126 369826 514324 454581 836581 989383 717415 088049 357180 527661 544825 262530 895748 488779 299633 327613 682314 528603 186773 780673 458286 456789 608098 646330 392860 673291 142931 268429 455714 773027 008289 332769 790302 864692 012339 156705 749706 753165 767840 324243 987425 205628 834908 872965 351936 722407 497808 560416 863895 515556 492763 007977 556487 670137 193012 987210 551124 843841 045445 637269 511657 336644 733060 370350 472052 658135 989057 860956 007133 332065 352079 887536 405994 580702 883193 694814 470144 / 3053 > 43264 [i]
- extracting embedded OOA [i] would yield OOA(43264, 654, S4, 5, 3052), but