Best Known (82, 82+∞, s)-Nets in Base 7
(82, 82+∞, 63)-Net over F7 — Constructive and digital
Digital (82, m, 63)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (82, 62)-sequence over F7, using
- Niederreiter–Xing sequence construction III based on function field F2/F7 with g(F2) = 21, N(F2) ≥ 2, and N(F2) + N2(F2) ≥ 63 from GarcÃa–Stichtenoth tower as constant field extension [i]
(82, 82+∞, 105)-Net over F7 — Digital
Digital (82, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (82, 104)-sequence over F7, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 43 and N(F) ≥ 105, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
(82, 82+∞, 515)-Net in Base 7 — Upper bound on s
There is no (82, m, 516)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (82, 1544, 516)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71544, 516, S7, 3, 1462), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 174303 820095 370881 301619 718784 069139 563210 679962 417537 832013 990637 022449 522000 609508 955144 121993 798070 480308 832288 297797 200653 113276 074268 604060 391665 546868 026023 385139 456155 051939 056768 733516 944756 733754 277908 389669 133837 210953 696209 584474 383116 087439 320269 953403 458190 146741 630232 504782 809723 846316 967001 240311 582923 200948 538545 484000 658686 976499 290699 235113 036294 828779 783396 541476 919613 669651 636788 985489 980472 681155 735578 233850 626302 842760 332856 633557 523078 804517 172203 586158 472323 475296 933628 919229 260420 118500 207003 575391 858064 496931 315885 265823 839906 453208 831716 902338 965778 040759 347077 097966 405493 476866 461495 840336 662567 987746 056121 922472 926927 040418 431179 566034 423586 912185 335210 601715 818001 905591 452137 790506 594051 613187 018143 114651 658589 942846 466207 374786 586651 192617 374373 152282 520698 916653 919019 459199 729409 769624 797043 191845 619416 251801 437470 229194 338589 815547 948067 965463 891208 531696 361972 760116 303866 677800 977699 102698 626636 818464 803978 728477 598476 802115 723241 764189 111733 710399 112440 547678 368921 244937 792152 048459 179007 641110 511860 961496 248782 829999 330040 489593 261150 985013 135407 960662 930129 343920 357110 347920 480546 896327 563052 942428 742072 746695 324394 356301 614538 803391 463999 583440 060926 829062 061333 090626 439219 828396 903221 068180 350556 566619 594741 354375 627145 626963 415565 371642 570135 685057 / 209 > 71544 [i]
- extracting embedded OOA [i] would yield OOA(71544, 516, S7, 3, 1462), but