Best Known (228−43, 228, s)-Nets in Base 3
(228−43, 228, 896)-Net over F3 — Constructive and digital
Digital (185, 228, 896)-net over F3, using
- t-expansion [i] based on digital (184, 228, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 57, 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, 57, 224)-net over F81, using
(228−43, 228, 3497)-Net over F3 — Digital
Digital (185, 228, 3497)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3228, 3497, F3, 43) (dual of [3497, 3269, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3228, 6566, F3, 43) (dual of [6566, 6338, 44]-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(33, 4, F3, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
- dual of repetition code with length 4 [i]
- oval in PG(2, 3) [i]
- construction X applied to C([0,21]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3228, 6566, F3, 43) (dual of [6566, 6338, 44]-code), using
(228−43, 228, 623593)-Net in Base 3 — Upper bound on s
There is no (185, 228, 623594)-net in base 3, because
- 1 times m-reduction [i] would yield (185, 227, 623594)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 025472 219145 249159 604448 403943 621723 096216 999426 037709 982092 862281 871888 705447 293508 567340 194140 282749 848717 > 3227 [i]