Best Known (109−24, 109, s)-Nets in Base 9
(109−24, 109, 4922)-Net over F9 — Constructive and digital
Digital (85, 109, 4922)-net over F9, using
- net defined by OOA [i] based on linear OOA(9109, 4922, F9, 24, 24) (dual of [(4922, 24), 118019, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(9109, 59064, F9, 24) (dual of [59064, 58955, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(9109, 59067, F9, 24) (dual of [59067, 58958, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(9106, 59049, F9, 24) (dual of [59049, 58943, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(991, 59049, F9, 21) (dual of [59049, 58958, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(93, 18, F9, 2) (dual of [18, 15, 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(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(9109, 59067, F9, 24) (dual of [59067, 58958, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(9109, 59064, F9, 24) (dual of [59064, 58955, 25]-code), using
(109−24, 109, 54717)-Net over F9 — Digital
Digital (85, 109, 54717)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9109, 54717, F9, 24) (dual of [54717, 54608, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(9109, 59067, F9, 24) (dual of [59067, 58958, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(9106, 59049, F9, 24) (dual of [59049, 58943, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(991, 59049, F9, 21) (dual of [59049, 58958, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(93, 18, F9, 2) (dual of [18, 15, 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(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(9109, 59067, F9, 24) (dual of [59067, 58958, 25]-code), using
(109−24, 109, large)-Net in Base 9 — Upper bound on s
There is no (85, 109, large)-net in base 9, because
- 22 times m-reduction [i] would yield (85, 87, large)-net in base 9, but