Best Known (107, 107+27, s)-Nets in Base 3
(107, 107+27, 640)-Net over F3 — Constructive and digital
Digital (107, 134, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (107, 136, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 34, 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, 34, 160)-net over F81, using
(107, 107+27, 1734)-Net over F3 — Digital
Digital (107, 134, 1734)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3134, 1734, F3, 27) (dual of [1734, 1600, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3134, 2216, F3, 27) (dual of [2216, 2082, 28]-code), using
- construction XX applied to Ce(27) ⊂ Ce(22) ⊂ Ce(21) [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(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(27) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3134, 2216, F3, 27) (dual of [2216, 2082, 28]-code), using
(107, 107+27, 215609)-Net in Base 3 — Upper bound on s
There is no (107, 134, 215610)-net in base 3, because
- 1 times m-reduction [i] would yield (107, 133, 215610)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2865 052743 759499 754500 122508 832359 515283 256157 489903 408220 741501 > 3133 [i]