Best Known (251−181, 251, s)-Nets in Base 4
(251−181, 251, 66)-Net over F4 — Constructive and digital
Digital (70, 251, 66)-net over F4, using
- t-expansion [i] based on digital (49, 251, 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
(251−181, 251, 105)-Net over F4 — Digital
Digital (70, 251, 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
(251−181, 251, 393)-Net over F4 — Upper bound on s (digital)
There is no digital (70, 251, 394)-net over F4, because
- 1 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
(251−181, 251, 466)-Net in Base 4 — Upper bound on s
There is no (70, 251, 467)-net in base 4, because
- 1 times m-reduction [i] would yield (70, 250, 467)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3 364632 974372 419588 296342 857677 057295 494285 238220 853084 044824 410749 480316 238841 217891 356519 997498 531498 386529 414283 727025 992399 976175 513231 858846 586064 > 4250 [i]