Best Known (210−150, 210, s)-Nets in Base 4
(210−150, 210, 66)-Net over F4 — Constructive and digital
Digital (60, 210, 66)-net over F4, using
- t-expansion [i] based on digital (49, 210, 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
(210−150, 210, 91)-Net over F4 — Digital
Digital (60, 210, 91)-net over F4, using
- t-expansion [i] based on digital (50, 210, 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
(210−150, 210, 376)-Net over F4 — Upper bound on s (digital)
There is no digital (60, 210, 377)-net over F4, because
- 2 times m-reduction [i] would yield digital (60, 208, 377)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4208, 377, F4, 148) (dual of [377, 169, 149]-code), but
- residual code [i] would yield OA(460, 228, S4, 37), but
- the linear programming bound shows that M ≥ 4 063856 865702 461109 434988 542147 550126 468452 401248 290460 178687 206981 384116 633600 / 3 020261 554987 735388 398172 544670 898492 334641 > 460 [i]
- residual code [i] would yield OA(460, 228, S4, 37), but
- extracting embedded orthogonal array [i] would yield linear OA(4208, 377, F4, 148) (dual of [377, 169, 149]-code), but
(210−150, 210, 405)-Net in Base 4 — Upper bound on s
There is no (60, 210, 406)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2 830808 549286 913391 395370 313494 737398 777579 791652 487459 798090 847351 762568 028622 201687 674694 576631 894973 822324 689823 615861 736488 > 4210 [i]