Best Known (224−29, 224, s)-Nets in Base 3
(224−29, 224, 12658)-Net over F3 — Constructive and digital
Digital (195, 224, 12658)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-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 (0, 14, 4)-net over F3, using
(224−29, 224, 62513)-Net over F3 — Digital
Digital (195, 224, 62513)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3224, 62513, F3, 2, 29) (dual of [(62513, 2), 124802, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3224, 88602, F3, 2, 29) (dual of [(88602, 2), 176980, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3223, 88602, F3, 2, 29) (dual of [(88602, 2), 176981, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3221, 88601, F3, 2, 29) (dual of [(88601, 2), 176981, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3221, 177202, F3, 29) (dual of [177202, 176981, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [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(3166, 177147, F3, 23) (dual of [177147, 176981, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(28) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(3221, 177202, F3, 29) (dual of [177202, 176981, 30]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3221, 88601, F3, 2, 29) (dual of [(88601, 2), 176981, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3223, 88602, F3, 2, 29) (dual of [(88602, 2), 176981, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3224, 88602, F3, 2, 29) (dual of [(88602, 2), 176980, 30]-NRT-code), using
(224−29, 224, large)-Net in Base 3 — Upper bound on s
There is no (195, 224, large)-net in base 3, because
- 27 times m-reduction [i] would yield (195, 197, large)-net in base 3, but