Best Known (72, 72+24, s)-Nets in Base 9
(72, 72+24, 1094)-Net over F9 — Constructive and digital
Digital (72, 96, 1094)-net over F9, using
- net defined by OOA [i] based on linear OOA(996, 1094, F9, 24, 24) (dual of [(1094, 24), 26160, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(996, 13128, F9, 24) (dual of [13128, 13032, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(996, 13132, F9, 24) (dual of [13132, 13036, 25]-code), using
- trace code [i] based on linear OA(8148, 6566, F81, 24) (dual of [6566, 6518, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(8148, 6566, F81, 24) (dual of [6566, 6518, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(996, 13132, F9, 24) (dual of [13132, 13036, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(996, 13128, F9, 24) (dual of [13128, 13032, 25]-code), using
(72, 72+24, 13132)-Net over F9 — Digital
Digital (72, 96, 13132)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(996, 13132, F9, 24) (dual of [13132, 13036, 25]-code), using
- trace code [i] based on linear OA(8148, 6566, F81, 24) (dual of [6566, 6518, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(8148, 6566, F81, 24) (dual of [6566, 6518, 25]-code), using
(72, 72+24, large)-Net in Base 9 — Upper bound on s
There is no (72, 96, large)-net in base 9, because
- 22 times m-reduction [i] would yield (72, 74, large)-net in base 9, but