Best Known (18−6, 18, s)-Nets in Base 3
(18−6, 18, 84)-Net over F3 — Constructive and digital
Digital (12, 18, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 6, 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
(18−6, 18, 110)-Net over F3 — Digital
Digital (12, 18, 110)-net over F3, using
- net defined by OOA [i] based on linear OOA(318, 110, F3, 6, 6) (dual of [(110, 6), 642, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(318, 110, F3, 5, 6) (dual of [(110, 5), 532, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(318, 110, F3, 6) (dual of [110, 92, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(317, 108, F3, 6) (dual of [108, 91, 7]-code), using
- linear OA(317, 109, F3, 5) (dual of [109, 92, 6]-code), using Gilbert–Varšamov bound and bm = 317 > Vbs−1(k−1) = 87 402097 [i]
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X with Varšamov bound [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(318, 110, F3, 6) (dual of [110, 92, 7]-code), using
- appending kth column [i] based on linear OOA(318, 110, F3, 5, 6) (dual of [(110, 5), 532, 7]-NRT-code), using
(18−6, 18, 659)-Net in Base 3 — Upper bound on s
There is no (12, 18, 660)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 387 693681 > 318 [i]