Best Known (39, 39+25, s)-Nets in Base 4
(39, 39+25, 130)-Net over F4 — Constructive and digital
Digital (39, 64, 130)-net over F4, using
- 2 times m-reduction [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
(39, 39+25, 131)-Net over F4 — Digital
Digital (39, 64, 131)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(464, 131, F4, 2, 25) (dual of [(131, 2), 198, 26]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(462, 130, F4, 2, 25) (dual of [(130, 2), 198, 26]-NRT-code), using
- extracting embedded OOA [i] based on digital (37, 62, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 31, 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, 31, 65)-net over F16, using
- extracting embedded OOA [i] based on digital (37, 62, 130)-net over F4, using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(462, 130, F4, 2, 25) (dual of [(130, 2), 198, 26]-NRT-code), using
(39, 39+25, 2543)-Net in Base 4 — Upper bound on s
There is no (39, 64, 2544)-net in base 4, because
- 1 times m-reduction [i] would yield (39, 63, 2544)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 85 255083 066701 144383 910122 806699 365755 > 463 [i]