Best Known (106, 106+27, s)-Nets in Base 3
(106, 106+27, 640)-Net over F3 — Constructive and digital
Digital (106, 133, 640)-net over F3, using
- 31 times duplication [i] based on digital (105, 132, 640)-net over F3, using
- t-expansion [i] based on digital (104, 132, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 33, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 33, 160)-net over F81, using
- t-expansion [i] based on digital (104, 132, 640)-net over F3, using
(106, 106+27, 1659)-Net over F3 — Digital
Digital (106, 133, 1659)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3133, 1659, F3, 27) (dual of [1659, 1526, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3133, 2214, F3, 27) (dual of [2214, 2081, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3133, 2214, F3, 27) (dual of [2214, 2081, 28]-code), using
(106, 106+27, 198136)-Net in Base 3 — Upper bound on s
There is no (106, 133, 198137)-net in base 3, because
- 1 times m-reduction [i] would yield (106, 132, 198137)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 955 032977 592788 167387 847983 275140 220166 638208 473282 540468 721347 > 3132 [i]