Best Known (89, 119, s)-Nets in Base 9
(89, 119, 875)-Net over F9 — Constructive and digital
Digital (89, 119, 875)-net over F9, using
- 91 times duplication [i] based on digital (88, 118, 875)-net over F9, using
- net defined by OOA [i] based on linear OOA(9118, 875, F9, 30, 30) (dual of [(875, 30), 26132, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(9118, 13125, F9, 30) (dual of [13125, 13007, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(9118, 13126, F9, 30) (dual of [13126, 13008, 31]-code), using
- trace code [i] based on linear OA(8159, 6563, F81, 30) (dual of [6563, 6504, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- trace code [i] based on linear OA(8159, 6563, F81, 30) (dual of [6563, 6504, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(9118, 13126, F9, 30) (dual of [13126, 13008, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(9118, 13125, F9, 30) (dual of [13125, 13007, 31]-code), using
- net defined by OOA [i] based on linear OOA(9118, 875, F9, 30, 30) (dual of [(875, 30), 26132, 31]-NRT-code), using
(89, 119, 13128)-Net over F9 — Digital
Digital (89, 119, 13128)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9119, 13128, F9, 30) (dual of [13128, 13009, 31]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9118, 13126, F9, 30) (dual of [13126, 13008, 31]-code), using
- trace code [i] based on linear OA(8159, 6563, F81, 30) (dual of [6563, 6504, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- trace code [i] based on linear OA(8159, 6563, F81, 30) (dual of [6563, 6504, 31]-code), using
- linear OA(9118, 13127, F9, 29) (dual of [13127, 13009, 30]-code), using Gilbert–Varšamov bound and bm = 9118 > Vbs−1(k−1) = 1252 237184 700134 344262 244718 352291 006239 349389 094798 092128 265019 087380 740526 948721 017214 203539 303742 109645 581809 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9118, 13126, F9, 30) (dual of [13126, 13008, 31]-code), using
- construction X with Varšamov bound [i] based on
(89, 119, large)-Net in Base 9 — Upper bound on s
There is no (89, 119, large)-net in base 9, because
- 28 times m-reduction [i] would yield (89, 91, large)-net in base 9, but