Best Known (228−29, 228, s)-Nets in Base 3
(228−29, 228, 12666)-Net over F3 — Constructive and digital
Digital (199, 228, 12666)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 18, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (181, 210, 12654)-net over F3, using
- net defined by OOA [i] based on linear OOA(3210, 12654, F3, 29, 29) (dual of [(12654, 29), 366756, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3210, 177157, F3, 29) (dual of [177157, 176947, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 177158, F3, 29) (dual of [177158, 176948, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3210, 177147, F3, 29) (dual of [177147, 176937, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3210, 177158, F3, 29) (dual of [177158, 176948, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3210, 177157, F3, 29) (dual of [177157, 176947, 30]-code), using
- net defined by OOA [i] based on linear OOA(3210, 12654, F3, 29, 29) (dual of [(12654, 29), 366756, 30]-NRT-code), using
- digital (4, 18, 12)-net over F3, using
(228−29, 228, 74029)-Net over F3 — Digital
Digital (199, 228, 74029)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3228, 74029, F3, 2, 29) (dual of [(74029, 2), 147830, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3228, 88610, F3, 2, 29) (dual of [(88610, 2), 176992, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3226, 88609, F3, 2, 29) (dual of [(88609, 2), 176992, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3226, 177218, F3, 29) (dual of [177218, 176992, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(3210, 177147, F3, 29) (dual of [177147, 176937, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(316, 71, F3, 6) (dual of [71, 55, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 80, F3, 6) (dual of [80, 64, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(316, 80, F3, 6) (dual of [80, 64, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3226, 177218, F3, 29) (dual of [177218, 176992, 30]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3226, 88609, F3, 2, 29) (dual of [(88609, 2), 176992, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3228, 88610, F3, 2, 29) (dual of [(88610, 2), 176992, 30]-NRT-code), using
(228−29, 228, large)-Net in Base 3 — Upper bound on s
There is no (199, 228, large)-net in base 3, because
- 27 times m-reduction [i] would yield (199, 201, large)-net in base 3, but