Best Known (90, ∞, s)-Nets in Base 7
(90, ∞, 71)-Net over F7 — Constructive and digital
Digital (90, m, 71)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (90, 70)-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) ≥ 71 from GarcÃa–Stichtenoth tower as constant field extension [i]
(90, ∞, 105)-Net over F7 — Digital
Digital (90, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (90, 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
(90, ∞, 563)-Net in Base 7 — Upper bound on s
There is no (90, m, 564)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (90, 1688, 564)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71688, 564, S7, 3, 1598), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 19 483531 430617 045265 407698 989607 848906 344954 338808 044439 071993 777290 970727 486082 010919 776116 990613 810396 982671 753852 858594 966353 010323 498186 672723 872569 918874 743779 444608 983993 198738 768985 581697 799173 526858 175149 343648 416595 286427 576945 441768 988729 144298 720063 704919 454818 823560 447284 878094 333117 015093 574219 195562 287366 475011 416294 571600 155079 405261 458886 307822 433324 659915 116662 774625 199760 352195 542655 128424 941480 710743 878720 973963 614965 508036 902018 367595 023712 871978 234404 703664 322259 664160 287574 520468 868274 704561 765186 999255 399079 387936 486720 377939 888754 608538 228904 109105 839241 436173 994389 502084 646338 298986 057184 650673 793064 771956 516736 182987 134866 497157 512075 047867 084789 101448 927808 070565 642799 096476 825176 434709 168639 061356 139837 291347 241338 333694 498039 907600 243686 604685 545312 191870 108694 253873 981661 161805 183805 492945 179684 315763 843290 398050 867105 809537 570522 817124 768775 101399 919881 108405 281456 917374 599789 132355 584964 610745 789972 738962 551727 230818 005906 884075 046333 406631 442844 249680 786010 561216 471133 587124 400798 020529 240150 696762 133383 522055 985836 434609 546569 995921 693478 580618 085603 995060 466725 707678 579656 118764 810077 024039 984410 643930 284596 013746 612043 924207 718287 270745 003524 402226 638450 420488 645020 519930 377229 801149 926575 209052 556188 275792 832237 187139 954062 225706 627229 526717 358888 522000 976933 254813 060507 696270 947253 230110 115490 819323 159563 320147 578725 116551 238287 074369 886667 479087 864298 536928 359819 797381 / 533 > 71688 [i]
- extracting embedded OOA [i] would yield OOA(71688, 564, S7, 3, 1598), but