Best Known (51−8, 51, s)-Nets in Base 3
(51−8, 51, 14764)-Net over F3 — Constructive and digital
Digital (43, 51, 14764)-net over F3, using
- net defined by OOA [i] based on linear OOA(351, 14764, F3, 8, 8) (dual of [(14764, 8), 118061, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(351, 59056, F3, 8) (dual of [59056, 59005, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(351, 59059, F3, 8) (dual of [59059, 59008, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(30, 10, F3, 0) (dual of [10, 10, 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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(351, 59059, F3, 8) (dual of [59059, 59008, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(351, 59056, F3, 8) (dual of [59056, 59005, 9]-code), using
(51−8, 51, 29529)-Net over F3 — Digital
Digital (43, 51, 29529)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(351, 29529, F3, 2, 8) (dual of [(29529, 2), 59007, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(351, 59058, F3, 8) (dual of [59058, 59007, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(351, 59059, F3, 8) (dual of [59059, 59008, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(30, 10, F3, 0) (dual of [10, 10, 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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(351, 59059, F3, 8) (dual of [59059, 59008, 9]-code), using
- OOA 2-folding [i] based on linear OA(351, 59058, F3, 8) (dual of [59058, 59007, 9]-code), using
(51−8, 51, 1340656)-Net in Base 3 — Upper bound on s
There is no (43, 51, 1340657)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 153693 992246 171397 492177 > 351 [i]