Best Known (189, 189+19, s)-Nets in Base 3
(189, 189+19, 932150)-Net over F3 — Constructive and digital
Digital (189, 208, 932150)-net over F3, using
- (u, u+v)-construction [i] based on
- 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
- trace code for nets [i] based on digital (0, 9, 28)-net over F27, using
- digital (162, 181, 932066)-net over F3, using
- net defined by OOA [i] based on linear OOA(3181, 932066, F3, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3181, 8388595, F3, 19) (dual of [8388595, 8388414, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3181, 8388595, F3, 19) (dual of [8388595, 8388414, 20]-code), using
- net defined by OOA [i] based on linear OOA(3181, 932066, F3, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
- digital (18, 27, 84)-net over F3, using
(189, 189+19, 4194396)-Net over F3 — Digital
Digital (189, 208, 4194396)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3208, 4194396, F3, 2, 19) (dual of [(4194396, 2), 8388584, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(327, 95, F3, 2, 9) (dual of [(95, 2), 163, 10]-NRT-code), 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
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(327, 95, F3, 9) (dual of [95, 68, 10]-code), using
- linear OOA(3181, 4194301, F3, 2, 19) (dual of [(4194301, 2), 8388421, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3181, 8388602, F3, 19) (dual of [8388602, 8388421, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- OOA 2-folding [i] based on linear OA(3181, 8388602, F3, 19) (dual of [8388602, 8388421, 20]-code), using
- linear OOA(327, 95, F3, 2, 9) (dual of [(95, 2), 163, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(189, 189+19, large)-Net in Base 3 — Upper bound on s
There is no (189, 208, large)-net in base 3, because
- 17 times m-reduction [i] would yield (189, 191, large)-net in base 3, but