Best Known (27, 40, s)-Nets in Base 3
(27, 40, 84)-Net over F3 — Constructive and digital
Digital (27, 40, 84)-net over F3, using
- 31 times duplication [i] based on digital (26, 39, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 13, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- trace code for nets [i] based on digital (0, 13, 28)-net over F27, using
(27, 40, 112)-Net over F3 — Digital
Digital (27, 40, 112)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(340, 112, F3, 13) (dual of [112, 72, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(340, 121, F3, 13) (dual of [121, 81, 14]-code), using
(27, 40, 1884)-Net in Base 3 — Upper bound on s
There is no (27, 40, 1885)-net in base 3, because
- 1 times m-reduction [i] would yield (27, 39, 1885)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 057871 719789 909425 > 339 [i]