Best Known (78, 78+142, s)-Nets in Base 3
(78, 78+142, 53)-Net over F3 — Constructive and digital
Digital (78, 220, 53)-net over F3, using
- net from sequence [i] based on digital (78, 52)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 52)-sequence over F9, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 52)-sequence over F9, using
(78, 78+142, 84)-Net over F3 — Digital
Digital (78, 220, 84)-net over F3, using
- t-expansion [i] based on digital (71, 220, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(78, 78+142, 288)-Net over F3 — Upper bound on s (digital)
There is no digital (78, 220, 289)-net over F3, because
- 1 times m-reduction [i] would yield digital (78, 219, 289)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3219, 289, F3, 141) (dual of [289, 70, 142]-code), but
- residual code [i] would yield OA(378, 147, S3, 47), but
- the linear programming bound shows that M ≥ 21292 547237 575438 242512 669266 940060 451971 468779 248833 468537 309663 893076 696653 339349 817917 027577 203520 372491 082583 925297 451510 978338 282476 549918 134976 940889 209055 346348 745588 096715 754691 301525 257875 911582 357689 118128 433314 737598 411437 554189 263641 384634 527177 045885 891311 119097 026068 187320 176874 779658 145764 333897 395803 493033 457257 474519 / 1271 885262 303799 475893 884858 058479 948832 016245 246803 267135 060438 027067 745487 272455 788616 343534 036690 595794 606685 181933 661012 606570 480122 034970 163911 161949 762660 647337 037879 497577 097723 009884 230765 932551 298588 319891 466627 072686 352792 728508 787497 192132 439728 152584 637120 637934 462669 771364 423271 801600 > 378 [i]
- residual code [i] would yield OA(378, 147, S3, 47), but
- extracting embedded orthogonal array [i] would yield linear OA(3219, 289, F3, 141) (dual of [289, 70, 142]-code), but
(78, 78+142, 344)-Net in Base 3 — Upper bound on s
There is no (78, 220, 345)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1041 948072 813054 522797 329425 375705 298867 943894 681190 449201 398904 152074 500271 923044 778274 170717 467879 644651 > 3220 [i]