Best Known (208−150, 208, s)-Nets in Base 4
(208−150, 208, 66)-Net over F4 — Constructive and digital
Digital (58, 208, 66)-net over F4, using
- t-expansion [i] based on digital (49, 208, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(208−150, 208, 91)-Net over F4 — Digital
Digital (58, 208, 91)-net over F4, using
- t-expansion [i] based on digital (50, 208, 91)-net over F4, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(208−150, 208, 341)-Net over F4 — Upper bound on s (digital)
There is no digital (58, 208, 342)-net over F4, because
- 2 times m-reduction [i] would yield digital (58, 206, 342)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4206, 342, F4, 148) (dual of [342, 136, 149]-code), but
- residual code [i] would yield OA(458, 193, S4, 37), but
- the linear programming bound shows that M ≥ 161285 735386 400830 134445 178003 287186 762218 409717 533224 067961 483465 588736 000000 / 1 868030 896794 654733 447451 649501 557791 288931 > 458 [i]
- residual code [i] would yield OA(458, 193, S4, 37), but
- extracting embedded orthogonal array [i] would yield linear OA(4206, 342, F4, 148) (dual of [342, 136, 149]-code), but
(208−150, 208, 388)-Net in Base 4 — Upper bound on s
There is no (58, 208, 389)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 170315 259224 821412 221286 710444 305336 155538 531891 192890 514272 012295 331328 694708 673548 505822 423469 569624 064718 390893 642296 141504 > 4208 [i]