Best Known (214−28, 214, s)-Nets in Base 3
(214−28, 214, 12660)-Net over F3 — Constructive and digital
Digital (186, 214, 12660)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (171, 199, 12653)-net over F3, using
- net defined by OOA [i] based on linear OOA(3199, 12653, F3, 28, 28) (dual of [(12653, 28), 354085, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3199, 177142, F3, 28) (dual of [177142, 176943, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3199, 177142, F3, 28) (dual of [177142, 176943, 29]-code), using
- net defined by OOA [i] based on linear OOA(3199, 12653, F3, 28, 28) (dual of [(12653, 28), 354085, 29]-NRT-code), using
- digital (1, 15, 7)-net over F3, using
(214−28, 214, 59068)-Net over F3 — Digital
Digital (186, 214, 59068)-net over F3, using
- 32 times duplication [i] based on digital (184, 212, 59068)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 59068, F3, 3, 28) (dual of [(59068, 3), 176992, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3212, 177204, F3, 28) (dual of [177204, 176992, 29]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3210, 177202, F3, 28) (dual of [177202, 176992, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(311, 55, F3, 5) (dual of [55, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3210, 177202, F3, 28) (dual of [177202, 176992, 29]-code), using
- OOA 3-folding [i] based on linear OA(3212, 177204, F3, 28) (dual of [177204, 176992, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 59068, F3, 3, 28) (dual of [(59068, 3), 176992, 29]-NRT-code), using
(214−28, 214, large)-Net in Base 3 — Upper bound on s
There is no (186, 214, large)-net in base 3, because
- 26 times m-reduction [i] would yield (186, 188, large)-net in base 3, but