Best Known (18, 18+9, s)-Nets in Base 3
(18, 18+9, 84)-Net over F3 — Constructive and digital
Digital (18, 27, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 9, 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, 18+9, 95)-Net over F3 — Digital
Digital (18, 27, 95)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(327, 95, F3, 9) (dual of [95, 68, 10]-code), using
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(326, 92, F3, 9) (dual of [92, 66, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(325, 82, F3, 9) (dual of [82, 57, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 82 | 38−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(317, 82, F3, 7) (dual of [82, 65, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 82 | 38−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(39, 10, F3, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,3)), using
- dual of repetition code with length 10 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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 X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(326, 92, F3, 9) (dual of [92, 66, 10]-code), using
(18, 18+9, 1393)-Net in Base 3 — Upper bound on s
There is no (18, 27, 1394)-net in base 3, because
- 1 times m-reduction [i] would yield (18, 26, 1394)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 542795 606313 > 326 [i]