Best Known (210−27, 210, s)-Nets in Base 3
(210−27, 210, 13630)-Net over F3 — Constructive and digital
Digital (183, 210, 13630)-net over F3, using
- 32 times duplication [i] based on digital (181, 208, 13630)-net over F3, using
- net defined by OOA [i] based on linear OOA(3208, 13630, F3, 27, 27) (dual of [(13630, 27), 367802, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3208, 177191, F3, 27) (dual of [177191, 176983, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3208, 177200, F3, 27) (dual of [177200, 176992, 28]-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(39, 53, F3, 4) (dual of [53, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3208, 177200, F3, 27) (dual of [177200, 176992, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3208, 177191, F3, 27) (dual of [177191, 176983, 28]-code), using
- net defined by OOA [i] based on linear OOA(3208, 13630, F3, 27, 27) (dual of [(13630, 27), 367802, 28]-NRT-code), using
(210−27, 210, 66870)-Net over F3 — Digital
Digital (183, 210, 66870)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3210, 66870, F3, 2, 27) (dual of [(66870, 2), 133530, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3210, 88601, F3, 2, 27) (dual of [(88601, 2), 176992, 28]-NRT-code), using
- strength reduction [i] based on linear OOA(3210, 88601, F3, 2, 28) (dual of [(88601, 2), 176992, 29]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(3210, 177202, F3, 28) (dual of [177202, 176992, 29]-code), using
- strength reduction [i] based on linear OOA(3210, 88601, F3, 2, 28) (dual of [(88601, 2), 176992, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3210, 88601, F3, 2, 27) (dual of [(88601, 2), 176992, 28]-NRT-code), using
(210−27, 210, large)-Net in Base 3 — Upper bound on s
There is no (183, 210, large)-net in base 3, because
- 25 times m-reduction [i] would yield (183, 185, large)-net in base 3, but