Best Known (66−10, 66, s)-Nets in Base 3
(66−10, 66, 11814)-Net over F3 — Constructive and digital
Digital (56, 66, 11814)-net over F3, using
- 31 times duplication [i] based on digital (55, 65, 11814)-net over F3, using
- net defined by OOA [i] based on linear OOA(365, 11814, F3, 10, 10) (dual of [(11814, 10), 118075, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(365, 59070, F3, 10) (dual of [59070, 59005, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(365, 59073, F3, 10) (dual of [59073, 59008, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(365, 59073, F3, 10) (dual of [59073, 59008, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(365, 59070, F3, 10) (dual of [59070, 59005, 11]-code), using
- net defined by OOA [i] based on linear OOA(365, 11814, F3, 10, 10) (dual of [(11814, 10), 118075, 11]-NRT-code), using
(66−10, 66, 29537)-Net over F3 — Digital
Digital (56, 66, 29537)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(366, 29537, F3, 2, 10) (dual of [(29537, 2), 59008, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(366, 59074, F3, 10) (dual of [59074, 59008, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(365, 59073, F3, 10) (dual of [59073, 59008, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(365, 59073, F3, 10) (dual of [59073, 59008, 11]-code), using
- OOA 2-folding [i] based on linear OA(366, 59074, F3, 10) (dual of [59074, 59008, 11]-code), using
(66−10, 66, 2587057)-Net in Base 3 — Upper bound on s
There is no (56, 66, 2587058)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 30 903191 731803 015092 263528 869885 > 366 [i]