Best Known (15, 15+7, s)-Nets in Base 3
(15, 15+7, 84)-Net over F3 — Constructive and digital
Digital (15, 22, 84)-net over F3, using
- 31 times duplication [i] based on digital (14, 21, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 28)-net over F27, using
(15, 15+7, 128)-Net over F3 — Digital
Digital (15, 22, 128)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(322, 128, F3, 7) (dual of [128, 106, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(322, 249, F3, 7) (dual of [249, 227, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(321, 243, F3, 7) (dual of [243, 222, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(316, 243, F3, 5) (dual of [243, 227, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(6) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(322, 249, F3, 7) (dual of [249, 227, 8]-code), using
(15, 15+7, 1984)-Net in Base 3 — Upper bound on s
There is no (15, 22, 1985)-net in base 3, because
- 1 times m-reduction [i] would yield (15, 21, 1985)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10467 893531 > 321 [i]