Best Known (41, 41+28, s)-Nets in Base 8
(41, 41+28, 256)-Net over F8 — Constructive and digital
Digital (41, 69, 256)-net over F8, using
- 3 times m-reduction [i] based on digital (41, 72, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 36, 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, 36, 128)-net over F64, using
(41, 41+28, 300)-Net in Base 8 — Constructive
(41, 69, 300)-net in base 8, using
- 1 times m-reduction [i] based on (41, 70, 300)-net in base 8, using
- trace code for nets [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
- trace code for nets [i] based on (6, 35, 150)-net in base 64, using
(41, 41+28, 333)-Net over F8 — Digital
Digital (41, 69, 333)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(869, 333, F8, 28) (dual of [333, 264, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(869, 334, F8, 28) (dual of [334, 265, 29]-code), using
- 11 step Varšamov–Edel lengthening with (ri) = (1, 10 times 0) [i] based on linear OA(868, 322, F8, 28) (dual of [322, 254, 29]-code), using
- trace code [i] based on linear OA(6434, 161, F64, 28) (dual of [161, 127, 29]-code), using
- extended algebraic-geometric code AGe(F,132P) [i] based on function field F/F64 with g(F) = 6 and N(F) ≥ 161, using
- trace code [i] based on linear OA(6434, 161, F64, 28) (dual of [161, 127, 29]-code), using
- 11 step Varšamov–Edel lengthening with (ri) = (1, 10 times 0) [i] based on linear OA(868, 322, F8, 28) (dual of [322, 254, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(869, 334, F8, 28) (dual of [334, 265, 29]-code), using
(41, 41+28, 24386)-Net in Base 8 — Upper bound on s
There is no (41, 69, 24387)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 205 731174 095451 368851 792822 729561 448567 706954 894128 718539 500464 > 869 [i]