Best Known (39, 39+25, s)-Nets in Base 8
(39, 39+25, 354)-Net over F8 — Constructive and digital
Digital (39, 64, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 32, 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
(39, 39+25, 389)-Net over F8 — Digital
Digital (39, 64, 389)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(864, 389, F8, 25) (dual of [389, 325, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(864, 511, F8, 25) (dual of [511, 447, 26]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(864, 511, F8, 25) (dual of [511, 447, 26]-code), using
(39, 39+25, 41630)-Net in Base 8 — Upper bound on s
There is no (39, 64, 41631)-net in base 8, because
- 1 times m-reduction [i] would yield (39, 63, 41631)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 784 774018 764018 373425 145993 938002 570702 301838 379561 654649 > 863 [i]