Best Known (69, 89, s)-Nets in Base 9
(69, 89, 5906)-Net over F9 — Constructive and digital
Digital (69, 89, 5906)-net over F9, using
- net defined by OOA [i] based on linear OOA(989, 5906, F9, 20, 20) (dual of [(5906, 20), 118031, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(989, 59060, F9, 20) (dual of [59060, 58971, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(989, 59062, F9, 20) (dual of [59062, 58973, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(986, 59049, F9, 20) (dual of [59049, 58963, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(976, 59049, F9, 17) (dual of [59049, 58973, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(93, 13, F9, 2) (dual of [13, 10, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(989, 59062, F9, 20) (dual of [59062, 58973, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(989, 59060, F9, 20) (dual of [59060, 58971, 21]-code), using
(69, 89, 43664)-Net over F9 — Digital
Digital (69, 89, 43664)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(989, 43664, F9, 20) (dual of [43664, 43575, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(989, 59062, F9, 20) (dual of [59062, 58973, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(986, 59049, F9, 20) (dual of [59049, 58963, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(976, 59049, F9, 17) (dual of [59049, 58973, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(93, 13, F9, 2) (dual of [13, 10, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(989, 59062, F9, 20) (dual of [59062, 58973, 21]-code), using
(69, 89, large)-Net in Base 9 — Upper bound on s
There is no (69, 89, large)-net in base 9, because
- 18 times m-reduction [i] would yield (69, 71, large)-net in base 9, but