Best Known (222−29, 222, s)-Nets in Base 3
(222−29, 222, 12657)-Net over F3 — Constructive and digital
Digital (193, 222, 12657)-net over F3, using
- 31 times duplication [i] based on digital (192, 221, 12657)-net over F3, using
- net defined by OOA [i] based on linear OOA(3221, 12657, F3, 29, 29) (dual of [(12657, 29), 366832, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3221, 177199, F3, 29) (dual of [177199, 176978, 30]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3221, 177202, F3, 29) (dual of [177202, 176981, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3221, 177199, F3, 29) (dual of [177199, 176978, 30]-code), using
- net defined by OOA [i] based on linear OOA(3221, 12657, F3, 29, 29) (dual of [(12657, 29), 366832, 30]-NRT-code), using
(222−29, 222, 59067)-Net over F3 — Digital
Digital (193, 222, 59067)-net over F3, using
- 31 times duplication [i] based on digital (192, 221, 59067)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3221, 59067, F3, 3, 29) (dual of [(59067, 3), 176980, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3221, 177201, F3, 29) (dual of [177201, 176980, 30]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3221, 177202, F3, 29) (dual of [177202, 176981, 30]-code), using
- OOA 3-folding [i] based on linear OA(3221, 177201, F3, 29) (dual of [177201, 176980, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3221, 59067, F3, 3, 29) (dual of [(59067, 3), 176980, 30]-NRT-code), using
(222−29, 222, large)-Net in Base 3 — Upper bound on s
There is no (193, 222, large)-net in base 3, because
- 27 times m-reduction [i] would yield (193, 195, large)-net in base 3, but