Best Known (70−27, 70, s)-Nets in Base 8
(70−27, 70, 354)-Net over F8 — Constructive and digital
Digital (43, 70, 354)-net over F8, using
- 2 times m-reduction [i] based on digital (43, 72, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 36, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 36, 177)-net over F64, using
(70−27, 70, 384)-Net in Base 8 — Constructive
(43, 70, 384)-net in base 8, using
- trace code for nets [i] based on (8, 35, 192)-net in base 64, using
- base change [i] based on digital (3, 30, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 30, 192)-net over F128, using
(70−27, 70, 438)-Net over F8 — Digital
Digital (43, 70, 438)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(870, 438, F8, 27) (dual of [438, 368, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(870, 511, F8, 27) (dual of [511, 441, 28]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(870, 511, F8, 27) (dual of [511, 441, 28]-code), using
(70−27, 70, 50299)-Net in Base 8 — Upper bound on s
There is no (43, 70, 50300)-net in base 8, because
- 1 times m-reduction [i] would yield (43, 69, 50300)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 205 712085 352008 522522 613714 603536 634662 967886 735241 651773 598636 > 869 [i]