Best Known (40−21, 40, s)-Nets in Base 27
(40−21, 40, 132)-Net over F27 — Constructive and digital
Digital (19, 40, 132)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (5, 26, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 14, 64)-net over F27, using
(40−21, 40, 172)-Net in Base 27 — Constructive
(19, 40, 172)-net in base 27, using
- 8 times m-reduction [i] based on (19, 48, 172)-net in base 27, using
- base change [i] based on digital (7, 36, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 36, 172)-net over F81, using
(40−21, 40, 256)-Net over F27 — Digital
Digital (19, 40, 256)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2740, 256, F27, 21) (dual of [256, 216, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2740, 364, F27, 21) (dual of [364, 324, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 364 | 272−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(2740, 364, F27, 21) (dual of [364, 324, 22]-code), using
(40−21, 40, 66571)-Net in Base 27 — Upper bound on s
There is no (19, 40, 66572)-net in base 27, because
- 1 times m-reduction [i] would yield (19, 39, 66572)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 66 558746 017838 387426 217460 098923 454489 925601 494719 590761 > 2739 [i]