Best Known (27, 36, s)-Nets in Base 3
(27, 36, 328)-Net over F3 — Constructive and digital
Digital (27, 36, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 9, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
(27, 36, 405)-Net over F3 — Digital
Digital (27, 36, 405)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(336, 405, F3, 9) (dual of [405, 369, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(336, 728, F3, 9) (dual of [728, 692, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(336, 728, F3, 9) (dual of [728, 692, 10]-code), using
(27, 36, 16547)-Net in Base 3 — Upper bound on s
There is no (27, 36, 16548)-net in base 3, because
- 1 times m-reduction [i] would yield (27, 35, 16548)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 50033 184071 190209 > 335 [i]