Best Known (186, 229, s)-Nets in Base 3
(186, 229, 896)-Net over F3 — Constructive and digital
Digital (186, 229, 896)-net over F3, using
- 31 times duplication [i] based on 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
- t-expansion [i] based on digital (184, 228, 896)-net over F3, using
(186, 229, 3593)-Net over F3 — Digital
Digital (186, 229, 3593)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3229, 3593, F3, 43) (dual of [3593, 3364, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 6572, F3, 43) (dual of [6572, 6343, 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(34, 10, F3, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), 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, 6572, F3, 43) (dual of [6572, 6343, 44]-code), using
(186, 229, 657086)-Net in Base 3 — Upper bound on s
There is no (186, 229, 657087)-net in base 3, because
- 1 times m-reduction [i] would yield (186, 228, 657087)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 076463 588670 317495 222254 312865 304325 411922 531313 465018 492040 538502 410705 190762 586704 146698 129672 541741 369175 > 3228 [i]