Best Known (104, ∞, s)-Nets in Base 7
(104, ∞, 85)-Net over F7 — Constructive and digital
Digital (104, m, 85)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (104, 84)-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) ≥ 85 from GarcÃa–Stichtenoth tower as constant field extension [i]
(104, ∞, 105)-Net over F7 — Digital
Digital (104, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (104, 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
(104, ∞, 648)-Net in Base 7 — Upper bound on s
There is no (104, m, 649)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (104, 1943, 649)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71943, 649, S7, 3, 1839), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 5 418918 855864 120242 503921 565941 618895 115798 105432 583872 491124 159055 715815 620132 944028 165575 839879 778973 527425 417749 164004 086027 995360 795022 904476 330759 891258 978581 375952 702393 184392 897041 290336 543272 684777 629759 363986 152188 976882 174661 412857 718783 851103 870961 126888 831787 917429 355419 746752 257274 721066 090385 114622 106532 009106 687174 624764 172037 265991 417220 697483 705254 659379 126360 759865 719428 109355 484235 702873 521720 232933 838291 855673 287780 086837 670612 049340 344014 828876 485404 720930 216649 474126 223994 153395 832676 560167 183726 359134 487429 222758 119998 469661 124492 717909 905801 553044 998956 086301 459860 983802 379386 612022 763416 326993 748585 386597 500067 583439 844859 319764 806352 689920 020280 244164 177827 403417 109062 169963 852036 795164 902786 110945 514226 619703 175594 465419 563175 603652 926642 088239 266262 212287 472051 554566 211238 041023 663393 727593 859460 059019 328581 638121 593798 327857 213432 946415 427899 546367 286848 827847 868979 559897 668333 845628 384699 714642 237458 919241 205063 952003 722905 534666 443991 953957 624741 470045 052314 039641 384017 413526 987985 393602 254174 229170 734105 121777 766348 567686 627939 870603 510746 429175 907554 759191 119735 167652 640310 541944 893327 883489 824656 786292 803752 546198 461681 841261 726277 826516 450585 689739 701083 780889 254746 802209 472806 708391 814276 116738 673806 949159 657226 968358 182032 260249 023601 492351 137424 545873 790666 499633 065378 968749 858233 827348 394066 514250 544003 871273 239088 030784 914475 189589 303502 335002 911328 716271 402043 758296 633047 147301 416594 932141 579947 321258 531847 095342 502902 764687 285331 603820 895867 395067 612437 699052 263876 435992 573494 091185 744383 190807 566300 422247 329387 652233 916570 638269 657492 177012 548679 853565 630630 240637 198635 547273 / 460 > 71943 [i]
- extracting embedded OOA [i] would yield OOA(71943, 649, S7, 3, 1839), but