Best Known (219−81, 219, s)-Nets in Base 4
(219−81, 219, 147)-Net over F4 — Constructive and digital
Digital (138, 219, 147)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 45, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (93, 174, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 87, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 87, 65)-net over F16, using
- digital (5, 45, 17)-net over F4, using
(219−81, 219, 152)-Net in Base 4 — Constructive
(138, 219, 152)-net in base 4, using
- t-expansion [i] based on (137, 219, 152)-net in base 4, using
- 1 times m-reduction [i] based on (137, 220, 152)-net in base 4, using
- trace code for nets [i] based on (27, 110, 76)-net in base 16, using
- base change [i] based on digital (5, 88, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- base change [i] based on digital (5, 88, 76)-net over F32, using
- trace code for nets [i] based on (27, 110, 76)-net in base 16, using
- 1 times m-reduction [i] based on (137, 220, 152)-net in base 4, using
(219−81, 219, 413)-Net over F4 — Digital
Digital (138, 219, 413)-net over F4, using
(219−81, 219, 10010)-Net in Base 4 — Upper bound on s
There is no (138, 219, 10011)-net in base 4, because
- 1 times m-reduction [i] would yield (138, 218, 10011)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 177465 514459 466879 262784 888842 180152 591338 603607 723969 377008 205199 767324 890820 026196 755389 280092 000868 292014 210757 407587 085212 598760 > 4218 [i]