Best Known (47−15, 47, s)-Nets in Base 3
(47−15, 47, 84)-Net over F3 — Constructive and digital
Digital (32, 47, 84)-net over F3, using
- 1 times m-reduction [i] based on digital (32, 48, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 28)-net over F27, using
(47−15, 47, 128)-Net over F3 — Digital
Digital (32, 47, 128)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(347, 128, F3, 15) (dual of [128, 81, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(346, 122, F3, 16) (dual of [122, 76, 17]-code), using an extension Ce(15) of the narrow-sense BCH-code C(I) with length 121 | 35−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(341, 122, F3, 13) (dual of [122, 81, 14]-code), using an extension Ce(12) of the narrow-sense BCH-code C(I) with length 121 | 35−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(31, 6, F3, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
(47−15, 47, 2301)-Net in Base 3 — Upper bound on s
There is no (32, 47, 2302)-net in base 3, because
- 1 times m-reduction [i] would yield (32, 46, 2302)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8873 163268 115993 312313 > 346 [i]