Best Known (41, 41+27, s)-Nets in Base 4
(41, 41+27, 130)-Net over F4 — Constructive and digital
Digital (41, 68, 130)-net over F4, using
- 2 times m-reduction [i] based on digital (41, 70, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 35, 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, 35, 65)-net over F16, using
(41, 41+27, 131)-Net over F4 — Digital
Digital (41, 68, 131)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(468, 131, F4, 2, 27) (dual of [(131, 2), 194, 28]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(466, 130, F4, 2, 27) (dual of [(130, 2), 194, 28]-NRT-code), using
- extracting embedded OOA [i] based on digital (39, 66, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 33, 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, 33, 65)-net over F16, using
- extracting embedded OOA [i] based on digital (39, 66, 130)-net over F4, using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(466, 130, F4, 2, 27) (dual of [(130, 2), 194, 28]-NRT-code), using
(41, 41+27, 2384)-Net in Base 4 — Upper bound on s
There is no (41, 68, 2385)-net in base 4, because
- 1 times m-reduction [i] would yield (41, 67, 2385)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 21878 247806 168594 327974 422392 206498 131520 > 467 [i]