Best Known (103, ∞, s)-Nets in Base 7
(103, ∞, 84)-Net over F7 — Constructive and digital
Digital (103, m, 84)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (103, 83)-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) ≥ 84 from GarcÃa–Stichtenoth tower as constant field extension [i]
(103, ∞, 105)-Net over F7 — Digital
Digital (103, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (103, 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
(103, ∞, 642)-Net in Base 7 — Upper bound on s
There is no (103, m, 643)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (103, 1925, 643)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71925, 643, S7, 3, 1822), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 13356 500594 513822 353699 603971 313407 751298 292945 728951 567878 771425 559143 968237 671143 811351 070518 494434 851059 168500 494660 982057 460626 308545 685283 154788 449360 885922 603117 079550 622307 028114 244608 275792 037056 439898 514255 857292 835258 767877 530125 325500 027995 707909 013012 610653 836332 926229 102494 219944 244651 528862 173820 789280 331038 889285 373620 072209 259554 335709 213488 341085 926943 852258 268278 571682 670877 879878 516371 391405 390727 761383 279452 919956 281266 229636 304104 932041 435445 145790 359634 225378 118167 578572 197339 278305 167794 422064 615974 435716 749313 912936 319707 613438 031319 051759 566187 561798 392869 172824 140471 294948 848902 428728 160473 240389 952870 644959 431354 457997 634023 410314 974937 714166 118190 936490 068552 853949 468397 732709 050179 899529 023258 542049 878466 037816 392014 030630 138179 829270 093594 995828 301926 863431 638248 607836 111668 157378 960900 999821 352825 903991 590229 893605 401053 577685 116927 364093 774081 948502 255204 027670 411829 953360 588659 264994 216187 566107 483813 079045 228246 258421 023802 543552 794575 976829 579445 029019 964173 609810 317922 038995 601872 001500 495722 881262 277579 614337 425640 669619 575989 469593 576705 385527 907630 432040 261565 805015 791706 126496 474252 655261 994169 501796 419803 167476 573576 894700 481851 464525 275761 880092 409546 095414 381254 434715 986397 737398 068109 873348 225007 196006 987973 952155 993889 627270 604406 335420 610030 924113 718713 730088 477482 922402 330326 707004 299870 582001 920034 245025 094166 788818 529382 391497 922483 138366 472769 506514 044542 253634 110799 487371 843951 911213 897883 232233 172419 560400 664777 323920 340822 904858 046583 485613 197748 295548 909675 556072 220545 683153 642049 037841 008330 115507 095815 932717 633177 559732 319338 972296 924931 250314 535157 / 1823 > 71925 [i]
- extracting embedded OOA [i] would yield OOA(71925, 643, S7, 3, 1822), but