Best Known (136−99, 136, s)-Nets in Base 4
(136−99, 136, 56)-Net over F4 — Constructive and digital
Digital (37, 136, 56)-net over F4, using
- t-expansion [i] based on digital (33, 136, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the 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) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(136−99, 136, 66)-Net over F4 — Digital
Digital (37, 136, 66)-net over F4, using
- net from sequence [i] based on digital (37, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 37 and N(F) ≥ 66, using
(136−99, 136, 166)-Net over F4 — Upper bound on s (digital)
There is no digital (37, 136, 167)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4136, 167, F4, 99) (dual of [167, 31, 100]-code), but
- construction Y1 [i] would yield
- OA(4135, 149, S4, 99), but
- the linear programming bound shows that M ≥ 2 040610 487096 181427 758937 498317 933435 839690 422249 792187 355243 784188 300883 648114 289820 041216 / 970 822125 > 4135 [i]
- OA(431, 167, S4, 18), but
- the linear programming bound shows that M ≥ 10 709160 698850 412098 268143 898591 232000 / 2 251884 149854 892341 > 431 [i]
- OA(4135, 149, S4, 99), but
- construction Y1 [i] would yield
(136−99, 136, 252)-Net in Base 4 — Upper bound on s
There is no (37, 136, 253)-net in base 4, because
- 1 times m-reduction [i] would yield (37, 135, 253)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2240 390062 880074 181404 542300 298707 874496 700247 513298 053192 489677 380354 137105 381864 > 4135 [i]