Best Known (68, 68+∞, s)-Nets in Base 5
(68, 68+∞, 82)-Net over F5 — Constructive and digital
Digital (68, m, 82)-net over F5 for arbitrarily large m, using
- net from sequence [i] based on digital (68, 81)-sequence over F5, using
- t-expansion [i] based on digital (48, 81)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 48 and N(F) ≥ 82, using
- t-expansion [i] based on digital (48, 81)-sequence over F5, using
(68, 68+∞, 132)-Net over F5 — Digital
Digital (68, m, 132)-net over F5 for arbitrarily large m, using
- net from sequence [i] based on digital (68, 131)-sequence over F5, using
- t-expansion [i] based on digital (67, 131)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 67 and N(F) ≥ 132, using
- t-expansion [i] based on digital (67, 131)-sequence over F5, using
(68, 68+∞, 289)-Net in Base 5 — Upper bound on s
There is no (68, m, 290)-net in base 5 for arbitrarily large m, because
- m-reduction [i] would yield (68, 1155, 290)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(51155, 290, S5, 4, 1087), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 153255 571338 138579 978106 746738 961803 040358 946436 722755 214687 793241 883301 866761 933089 582489 152758 899732 069123 372228 191912 515637 045198 962296 678733 751494 828266 572718 238053 705848 706835 926657 033414 816784 282035 690943 563662 878011 330205 152982 199620 210994 584268 766866 023734 917738 742339 269570 419393 558501 387448 781440 872736 505781 678395 125327 417441 204148 086841 787194 449667 583168 134275 060594 847459 564109 432572 446324 978459 741213 810260 836188 317919 179681 716832 519019 134911 899333 698347 335610 148994 728848 971235 883655 732729 542446 536726 271583 205611 102622 407317 223077 462584 151735 442980 939765 985314 413146 277113 282314 055848 586910 088447 831491 309727 118792 563255 711468 533768 410520 124344 846441 514781 018526 675147 387671 658706 285474 093355 609036 280068 859308 023984 121484 014546 299040 050306 958330 120373 455718 318922 436083 084903 657436 370849 609375 / 68 > 51155 [i]
- extracting embedded OOA [i] would yield OOA(51155, 290, S5, 4, 1087), but