Best Known (85, 85+∞, s)-Nets in Base 7
(85, 85+∞, 66)-Net over F7 — Constructive and digital
Digital (85, m, 66)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (85, 65)-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) ≥ 66 from GarcÃa–Stichtenoth tower as constant field extension [i]
(85, 85+∞, 105)-Net over F7 — Digital
Digital (85, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (85, 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
(85, 85+∞, 533)-Net in Base 7 — Upper bound on s
There is no (85, m, 534)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (85, 1598, 534)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71598, 534, S7, 3, 1513), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2603 572668 187659 569201 576681 228125 941635 075152 603036 807941 024351 155424 415813 534955 700463 397256 909196 202313 042807 764637 977917 702826 682588 616608 085653 240053 635090 510793 792837 537787 135084 612026 534693 940197 601463 265048 407749 819461 815288 049649 853473 916402 357731 203339 479071 726729 176142 600410 570008 592316 200046 800891 985609 942061 029563 443996 770003 831881 655410 372036 608765 957008 555966 832655 841569 689336 112037 300298 868634 267065 696213 827577 623206 639572 732404 022563 033804 191408 620273 956128 099165 367266 755291 375589 766893 625051 064586 737411 400678 380175 997602 972806 945114 158080 976719 522678 125481 327330 387862 627229 234816 406262 402059 745980 572210 585913 837036 390158 963147 169478 096333 527522 028531 082214 844874 759065 939846 597205 378353 125647 535802 055536 403802 679885 229521 699029 396542 751261 484548 193925 249787 345962 264922 568475 258473 124381 162724 584324 014442 086905 003411 827831 543762 402254 430197 667414 848436 420726 045414 727237 379131 811207 879615 150153 608627 616718 748707 371791 848504 907939 601035 270595 574261 199447 488418 816376 387004 815636 538339 456203 391265 418645 316829 453100 766920 475012 059051 431292 457064 393588 760929 835540 727117 103062 892047 762697 168065 708290 659891 884816 312798 471651 032528 102555 898433 467377 800167 441916 686321 589348 855534 767184 390646 025968 717738 981056 835883 123518 097011 410561 679276 553889 175157 877700 597577 394852 849786 989548 479833 385088 311282 083536 926116 942006 214545 417161 / 757 > 71598 [i]
- extracting embedded OOA [i] would yield OOA(71598, 534, S7, 3, 1513), but