Best Known (187, 229, s)-Nets in Base 3
(187, 229, 896)-Net over F3 — Constructive and digital
Digital (187, 229, 896)-net over F3, using
- 3 times m-reduction [i] based on digital (187, 232, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 58, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 58, 224)-net over F81, using
(187, 229, 4096)-Net over F3 — Digital
Digital (187, 229, 4096)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3229, 4096, F3, 42) (dual of [4096, 3867, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 6582, F3, 42) (dual of [6582, 6353, 43]-code), using
- construction X applied to C([0,21]) ⊂ C([0,19]) [i] based on
- linear OA(3225, 6562, F3, 43) (dual of [6562, 6337, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to C([0,21]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3229, 6582, F3, 42) (dual of [6582, 6353, 43]-code), using
(187, 229, 692378)-Net in Base 3 — Upper bound on s
There is no (187, 229, 692379)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 18 229621 491711 017689 100092 074897 259723 285714 510266 977044 250676 143222 291511 158884 621915 574750 089372 619053 625679 > 3229 [i]