Best Known (243, 243+∞, s)-Nets in Base 4
(243, 243+∞, 258)-Net over F4 — Constructive and digital
Digital (243, m, 258)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (243, 257)-sequence over F4, using
- t-expansion [i] based on digital (225, 257)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 225 and N(F) ≥ 258, using
- T8 from the second 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) = 225 and N(F) ≥ 258, using
- t-expansion [i] based on digital (225, 257)-sequence over F4, using
(243, 243+∞, 301)-Net over F4 — Digital
Digital (243, m, 301)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (243, 300)-sequence over F4, using
- t-expansion [i] based on digital (234, 300)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 234 and N(F) ≥ 301, using
- t-expansion [i] based on digital (234, 300)-sequence over F4, using
(243, 243+∞, 747)-Net in Base 4 — Upper bound on s
There is no (243, m, 748)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (243, 3734, 748)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(43734, 748, S4, 5, 3491), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 532601 702902 924202 281605 984074 579592 223055 459030 293801 923645 255100 587892 276188 097040 204063 815097 869704 600340 593819 793826 961157 476582 659833 956860 789094 073199 627639 587571 116904 266365 638933 554379 760913 751508 260388 681304 469002 600113 477679 128102 519061 807225 410430 291962 510459 226155 958990 016791 055079 598651 130275 676484 401380 571969 516445 030516 502294 777049 385379 948355 538685 805511 268624 082576 876461 284486 927344 510650 902979 605907 367730 621103 482912 783937 112549 315109 040332 152195 073218 958134 779207 785340 146945 524795 887757 855022 019094 027362 821613 787497 671852 934107 448534 672639 069378 991202 300692 344468 279169 212890 386051 421676 100632 641486 959902 025126 186103 087574 895567 472537 802525 574202 557392 576016 401992 956889 970436 537641 566161 118583 097020 679286 986694 568299 980683 609761 769954 328733 056019 143806 352594 079871 702350 649319 912796 698911 932600 811218 445301 190418 097504 750068 385044 793429 738702 581328 218904 612033 179159 485902 867048 148374 373238 942737 984537 624943 739163 374938 511481 573115 227122 641684 404687 640637 468150 268790 089123 310096 781190 343687 882197 529814 712697 546443 338973 062749 143987 721675 461205 876373 402518 295589 244695 828141 122608 156908 500478 993168 421759 841187 778216 350635 549940 337918 805973 891418 458597 131719 248582 623162 157504 670389 628293 214046 906909 378658 919915 091851 302158 463331 720452 410109 032107 956801 769073 560781 719586 835181 191201 139927 753756 203135 359493 934064 055676 544993 852189 762785 244096 142168 328734 803401 704650 440450 551767 221120 579210 594589 882502 038569 744020 497907 809795 345943 722451 931994 631173 205623 351548 816158 844219 245601 489994 094407 763802 257707 850121 126971 683625 100887 117696 618802 940720 062201 669285 233441 605321 344556 223101 919115 491356 270788 216262 980152 525277 286365 291237 067938 562160 381790 529653 620687 794982 109149 306924 911009 894351 423315 845908 080616 511166 751320 954823 075632 708736 847027 718738 144709 623065 992724 413736 441328 867008 359150 971854 447718 686075 765792 941254 769130 336780 581188 539260 769472 524408 752186 747965 211057 402676 590446 651930 874159 353640 702399 465146 085881 426977 197324 425189 704077 060576 162848 188746 146777 099257 606194 414940 340549 431049 027403 445010 997916 118103 594048 457578 010143 274792 752469 692546 277294 809565 261418 151450 016630 382548 548017 562342 859676 024526 752613 378635 315998 734022 886032 875016 525444 389366 945003 778476 205344 325795 010628 691102 161797 382144 / 97 > 43734 [i]
- extracting embedded OOA [i] would yield OOA(43734, 748, S4, 5, 3491), but