Best Known (67−27, 67, s)-Nets in Base 8
(67−27, 67, 256)-Net over F8 — Constructive and digital
Digital (40, 67, 256)-net over F8, using
- 3 times m-reduction [i] based on digital (40, 70, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 35, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 35, 128)-net over F64, using
(67−27, 67, 300)-Net in Base 8 — Constructive
(40, 67, 300)-net in base 8, using
- 1 times m-reduction [i] based on (40, 68, 300)-net in base 8, using
- trace code for nets [i] based on (6, 34, 150)-net in base 64, using
- 1 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- base change [i] based on digital (1, 30, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 30, 150)-net over F128, using
- 1 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- trace code for nets [i] based on (6, 34, 150)-net in base 64, using
(67−27, 67, 336)-Net over F8 — Digital
Digital (40, 67, 336)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(867, 336, F8, 27) (dual of [336, 269, 28]-code), using
- 13 step Varšamov–Edel lengthening with (ri) = (1, 12 times 0) [i] based on linear OA(866, 322, F8, 27) (dual of [322, 256, 28]-code), using
- trace code [i] based on linear OA(6433, 161, F64, 27) (dual of [161, 128, 28]-code), using
- extended algebraic-geometric code AGe(F,133P) [i] based on function field F/F64 with g(F) = 6 and N(F) ≥ 161, using
- trace code [i] based on linear OA(6433, 161, F64, 27) (dual of [161, 128, 28]-code), using
- 13 step Varšamov–Edel lengthening with (ri) = (1, 12 times 0) [i] based on linear OA(866, 322, F8, 27) (dual of [322, 256, 28]-code), using
(67−27, 67, 31125)-Net in Base 8 — Upper bound on s
There is no (40, 67, 31126)-net in base 8, because
- 1 times m-reduction [i] would yield (40, 66, 31126)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 401811 310236 288782 043556 315399 009003 805156 318930 760956 194638 > 866 [i]