Best Known (84, 84+153, s)-Nets in Base 3
(84, 84+153, 59)-Net over F3 — Constructive and digital
Digital (84, 237, 59)-net over F3, using
- net from sequence [i] based on digital (84, 58)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 58)-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, 58)-sequence over F9, using
(84, 84+153, 84)-Net over F3 — Digital
Digital (84, 237, 84)-net over F3, using
- t-expansion [i] based on digital (71, 237, 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
(84, 84+153, 298)-Net over F3 — Upper bound on s (digital)
There is no digital (84, 237, 299)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3237, 299, F3, 153) (dual of [299, 62, 154]-code), but
- residual code [i] would yield OA(384, 145, S3, 51), but
- the linear programming bound shows that M ≥ 9326 522965 256564 011402 040940 711708 948592 367066 968406 717901 420916 222250 180273 247715 599586 772622 743299 930957 345787 791848 321048 478033 225610 993066 858007 366908 252164 894840 819421 014982 552403 762139 974174 388336 464326 612451 / 704454 170656 270579 836260 771554 871326 380655 415920 718317 470549 165243 209757 932338 987162 835617 965064 070982 145472 912111 399993 084597 556361 004087 761186 718070 628622 989614 325672 771840 > 384 [i]
- residual code [i] would yield OA(384, 145, S3, 51), but
(84, 84+153, 370)-Net in Base 3 — Upper bound on s
There is no (84, 237, 371)-net in base 3, because
- 1 times m-reduction [i] would yield (84, 236, 371)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 42016 842738 016132 342746 982998 395015 241445 330521 365272 593434 971435 351609 980430 360151 172640 723352 391363 176585 060553 > 3236 [i]