Best Known (253−183, 253, s)-Nets in Base 4
(253−183, 253, 66)-Net over F4 — Constructive and digital
Digital (70, 253, 66)-net over F4, using
- t-expansion [i] based on digital (49, 253, 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
(253−183, 253, 105)-Net over F4 — Digital
Digital (70, 253, 105)-net over F4, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 70 and N(F) ≥ 105, using
(253−183, 253, 393)-Net over F4 — Upper bound on s (digital)
There is no digital (70, 253, 394)-net over F4, because
- 3 times m-reduction [i] would yield digital (70, 250, 394)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4250, 394, F4, 180) (dual of [394, 144, 181]-code), but
- residual code [i] would yield OA(470, 213, S4, 45), but
- the linear programming bound shows that M ≥ 10138 160863 655691 762503 861603 731431 648994 530508 429756 680407 696662 337185 130982 211720 965164 892160 / 6819 841021 187779 545947 587667 623913 181856 482312 426343 > 470 [i]
- residual code [i] would yield OA(470, 213, S4, 45), but
- extracting embedded orthogonal array [i] would yield linear OA(4250, 394, F4, 180) (dual of [394, 144, 181]-code), but
(253−183, 253, 465)-Net in Base 4 — Upper bound on s
There is no (70, 253, 466)-net in base 4, because
- 1 times m-reduction [i] would yield (70, 252, 466)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 57 205214 733067 065835 257247 378536 388991 828470 763385 163486 843481 363315 370524 062227 085123 668243 910359 762082 949483 612876 554501 242543 835813 788207 711496 296256 > 4252 [i]