Best Known (25, 40, s)-Nets in Base 8
(25, 40, 256)-Net over F8 — Constructive and digital
Digital (25, 40, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 20, 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
(25, 40, 408)-Net over F8 — Digital
Digital (25, 40, 408)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(840, 408, F8, 15) (dual of [408, 368, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(840, 511, F8, 15) (dual of [511, 471, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(840, 511, F8, 15) (dual of [511, 471, 16]-code), using
(25, 40, 514)-Net in Base 8 — Constructive
(25, 40, 514)-net in base 8, using
- base change [i] based on digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 15, 257)-net over F256, using
(25, 40, 51914)-Net in Base 8 — Upper bound on s
There is no (25, 40, 51915)-net in base 8, because
- 1 times m-reduction [i] would yield (25, 39, 51915)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 166165 160244 849161 437599 503316 539512 > 839 [i]