Best Known (8, 8+12, s)-Nets in Base 16
(8, 8+12, 65)-Net over F16 — Constructive and digital
Digital (8, 20, 65)-net over F16, using
- t-expansion [i] based on digital (6, 20, 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
(8, 8+12, 66)-Net over F16 — Digital
Digital (8, 20, 66)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1620, 66, F16, 2, 12) (dual of [(66, 2), 112, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(1618, 65, F16, 2, 12) (dual of [(65, 2), 112, 13]-NRT-code), using
- extended algebraic-geometric NRT-code AGe(2;F,117P) [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- extended algebraic-geometric NRT-code AGe(2;F,117P) [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(1618, 65, F16, 2, 12) (dual of [(65, 2), 112, 13]-NRT-code), using
(8, 8+12, 80)-Net in Base 16 — Constructive
(8, 20, 80)-net in base 16, using
- 1 times m-reduction [i] based on (8, 21, 80)-net in base 16, using
- base change [i] based on digital (1, 14, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- base change [i] based on digital (1, 14, 80)-net over F64, using
(8, 8+12, 81)-Net in Base 16
(8, 20, 81)-net in base 16, using
- 1 times m-reduction [i] based on (8, 21, 81)-net in base 16, using
- base change [i] based on digital (1, 14, 81)-net over F64, using
- net from sequence [i] based on digital (1, 80)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 81, using
- net from sequence [i] based on digital (1, 80)-sequence over F64, using
- base change [i] based on digital (1, 14, 81)-net over F64, using
(8, 8+12, 2057)-Net in Base 16 — Upper bound on s
There is no (8, 20, 2058)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 1 212146 256864 075952 328596 > 1620 [i]