Best Known (105, ∞, s)-Nets in Base 7
(105, ∞, 86)-Net over F7 — Constructive and digital
Digital (105, m, 86)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (105, 85)-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) ≥ 86 from GarcÃa–Stichtenoth tower as constant field extension [i]
(105, ∞, 105)-Net over F7 — Digital
Digital (105, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (105, 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
(105, ∞, 654)-Net in Base 7 — Upper bound on s
There is no (105, m, 655)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (105, 1961, 655)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71961, 655, S7, 3, 1856), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 11725 361529 228208 034954 502665 800366 694116 443288 572008 413256 764184 453115 232582 918185 164493 174821 670438 171477 785568 020764 796557 440217 551456 012636 358455 168439 484999 173981 892596 775174 448546 599257 144760 292404 732179 582655 292490 895343 018937 536664 874992 954273 052894 098300 777884 681600 289204 775608 200134 716329 493779 189497 770449 076447 760968 992527 868458 425283 363551 705397 673346 979106 975153 688935 578138 733188 963253 708920 731864 112861 925222 953431 603419 069477 320883 505872 926541 753138 815885 031299 611883 640051 884267 546730 563346 985498 883614 056556 375802 921205 428667 944128 256538 007078 617693 177813 044492 556132 473004 836905 543120 160472 366078 806472 389736 332986 351792 052214 372223 286040 024765 773595 079069 754847 712016 529334 266997 996789 993507 524425 980350 635809 749167 844612 925461 524896 627629 310531 856291 520069 229263 333269 385905 003604 303466 098801 856522 945671 037969 800726 641960 600604 932208 168799 317448 291594 595732 841617 358221 160930 777943 195325 292323 981256 210867 958237 239152 206753 476442 218955 979900 318780 698439 234178 442583 996825 384549 381228 669102 976498 858410 335669 654290 501590 452477 096308 780920 937074 910595 668059 039098 302728 528014 951234 540815 477688 020813 210605 828348 734342 387726 476217 153899 981250 245663 676950 428185 702580 551411 647707 076468 175886 013182 071058 713331 545221 089097 551518 566678 974269 318215 538151 710692 565669 819187 611118 975138 981624 648292 558078 306118 036838 742469 866864 401616 232033 428259 310756 417332 155294 133675 149655 222937 644004 080576 729421 834475 167464 213027 406112 151207 245588 352104 380841 417710 431009 982268 893605 517711 561675 092200 891585 940898 123002 662785 611868 754543 669713 290290 485124 453838 461097 134730 727640 387943 153392 186020 595645 015844 423038 135410 449229 633702 380617 446626 720779 932753 / 619 > 71961 [i]
- extracting embedded OOA [i] would yield OOA(71961, 655, S7, 3, 1856), but